What's the question?

Recently, Dan Meyer issued the #anyqs challenge on his blog. Essentially, he is asking people to consider one photo or one minute of video that tells a compelling mathematical story that results in everyone wondering about the same question. If you find an interesting picture/video, then Tweet it using the "any questions" hashtag (#anyqs) and, to quote Dan, "I can promise you'll get a question from me, at the very least."


That same evening, I was at Michigan State University's Advance Degrees Graduation to watch my step-daughter's boyfriend receive his Ph.D. in Physics. He walked across the stage pretty early, which left me wondering what to do with the rest of my time. I remembered Dan's challenge and began looking through my pictures to see if any were appropriate. None elicited any compelling questions, so I put my phone away and looked down at the scene below me. That is when the question struck me. This is the picture that I sent out:

True to his word, Dan responded with his question:

And then, to my surprise, he threw it open to the Twitterverse:

Needless to say, the rest of my time was preoccupied responding to the many questions that my picture suggested:

Clearly, the picture needs work since so many different questions were asked. And no one came up with the original question I asked myself as I looked down on the throng, "Where's Waldo?" (Just kidding, but I couldn't resist.)

How will you supplement your curriculum? Part III

This series began with an overview of the Outside Resources section of the Professional Portfolio developed by each Teacher Assistant. In the second post, Brock, one of our TAs, shared his Outside Resources entry, and I admitted that I evaluated it incorrectly. The purpose of this, the final post in the series, is to explain what led to my error in evaluation and how the exit interview helped me to correct my misconception.

My original evaluation of Brock's work was Progressing. Here is how this level is described in the rubric: "The candidate collects outside resources but they are not related to the mathematics of the unit." Typically, TAs include an article/activity from NCTM that directly relates to the mathematical content of their unit. This is what I expected - my bias. 

When I read Brock's entry, I was surprised to find that he had included the NCTM Process Standards and Cambourne's Conditions of Learning as his Outside Resources. How did this relate to the mathematics found in his polynomial unit? I wrote a note reminding me to ask Brock about his choices and recorded "Progressing" in my NCATE spreadsheet.

About halfway through the exit interview, I questioned Brock about his Outside Resources. He responded that he thought the Process Standards and Conditions were important frameworks that should be implemented everyday in every lesson. My jaw dropped. How could I have missed this so completely? I had fallen into the old trap of judging instead of evaluating.

Based on my prior experiences with the portfolios, I was expecting TAs to produce a particular product - activities that would address some math content. Brock's entry was missing this product and so it was wrong. I had completely forgotten the fact that we had spent the semester discussing how the processes are as important as the content when it comes to math. I had completely forgotten to read Brock's entry with an open mind while using the questions from the Teaching-Learning Cycle to support my evaluation.

Fortunately, the exit interview provided me with the opportunity to be aware of my mistake, accept it, and make the necessary adjustment. Brock's entry was actually "Distinguished" given my new insight. He had gone beyond selecting a single activity that would enhance one topic in his unit. The Outside Resources he collected would impact every lesson, every day, regardless of what or who he was teaching. In his own words: "Including them in this portfolio is a way for me to have a constant reminder of them."

Sorry, Brock. Next time, I will be more open to alternative ideas and stick to our evaluation framework.  Next time, I'll do it better.

How will you supplement your curriculum? Part II

The previous post introduced the assessment used to evaluate Teacher Assistants’ ability to supplement their curriculum with outside resources. Typically, this entry includes an NCTM article related to the content addressed in a TA’s unit plan. Issues only arise when TAs select resources that are either unrelated to the mathematics in the unit or are not from NCTM.

(0) Unsatisfactory	(1) Progressing	(2) Proficient	(3) Distinguished
The candidate provides no outside resources from professional mathematics organizations.	The candidate collects outside resources but they are not related to the mathematics of the unit.	Appropriate resources are collected and the candidate explains how they relate to the content of his or her math unit.	The outside resources selected by the candidate either effectively extend students’ learning of a topic in the unit or provide scaffolding to students in need of support.

This year, one of our TAs, Brock Walsh, submitted the following entry. (He granted his permission for me to share this with you.) I was completely unprepared for this approach since it did not meet my experience or expectations. Consequently, I scored it as "Progressing" - surprise, I was wrong.

Outside Resources

The Process Standards from the NCTM website are a useful tool that I continuously find myself looking for and applying to my teaching philosophy. In particularly, I used these standards during one of my lessons this semester as I put myself at the front of the classroom to analyze a rational function problem on the white-board. I told the students that the purpose of this was for them to see what a good problem solver would look like so that they could reciprocate those ideas when they problem solved in the investigation. Showing the students my thought processes of problem solving, using reasoning and proof, and making connections were all things that I acted out in class for them to see. I then communicated to them what my thinking was and represented my work through graphs and charts. I had so much fun with this lesson and was able to improve on it teaching that lesson three times. I had an action plan set for that class as well looking to increase students’ engagement in problem solving and in the end, I felt that I did a very good job in my objective.

The Process Standards should always be a crucial framework to follow whenever I plan a lesson for my students. I want to make sure that when they are doing math, that they are able to somehow use these standards to show that they are mastering their work. My unit in particular needed a lot of connections to be made to understand new material. The students themselves needed to show me that they knew the material by communicating their thinking to me through formative assessments. We had projects where the students were able to represent their thinking on posters that were hung in the hallway. And lastly, they use reasoning during problem solving to work through problems and generate answers. My students will be successful when they are successfully using each of these standards appropriately.

Brian Cambourne’s Conditions of Learning is another great resource for really knowing how to engage my learners. Being an effective teacher means that I am reaching each student at the appropriate level that they should be learning. Being at Black River, we have high expectations for our students to be responsible for their own learning. In several lessons we allowed students to make approximations before we stepped in to demonstrate the correct way to problem solve. By doing that we gave them opportunity to achieve their own way of understanding the problem, which is powerful when it is accomplished. From the formative assessments, I was able to give students immediate response to their work and let them know how to make corrections to their understanding. All of these conditions truly make students progress towards better literacy of mathematics.

Cambourne’s conditions should be on my mind whenever I lesson plan. I need to know my students and present material to them in a way that they are engaged and are able to progressively develop their thinking. This is a framework that I should keep on hand and refer back to on occasion to make sure that I am reaching my student’s needs.

To be continued...

How will you supplement your curriculum? Part I

Last week 17 GVSU math majors completed their Secondary Teacher Assisting placement.  The final activity of the semester is an individual exit interview during which we meet with each teacher in training to go over his or her professional portfolio. Each portfolio includes 14 entries divided between three sections.

This series of posts focuses on the Outside Resources portion of the portfolio and what I learned about assessment and evaluation as a result of one Teacher Assistant’s entry in this category. Before we get to the TA’s work, however, here is some background information. The following details regarding the Outside Resources entry are given to the TAs at the start of the semester.

Objective: Select, use, and determine the suitability of the wide variety of available curricula and teaching materials for all students including those with special needs such as the gifted, challenged and speakers of other languages.

Task: Identify two outside resources that could be useful in the teaching of your unit.  At least one of these should be from the National Council of Teacher of Mathematics

Essential Questions: How does each resource fit into your unit? How does each resource aid in the planning, instruction or assessment of your unit?

Next, I want to share the rubric we use to evaluate our program for National Council for Accreditation of Teacher Education. This is based on TA’s work but it does not impact their grade. Consequently, we do not share this with the TAs.

Outside Resources: The candidate develops a collection of instructional resources (from NCTM) related to the topics of his or her unit that could strengthen the curriculum. An overview of the ideas along with how they might be implemented will accompany the resources

Indicator 8.1: Selects, uses, and determines the suitability of the wide variety of available mathematics curricula and teaching materials for all students including those with special needs such as the gifted, challenged and speakers of other languages.

Indicator 8.5: Participates in professional mathematics organizations and uses their print and on-line resources.

(0) Unsatisfactory	(1) Progressing	(2) Proficient	(3) Distinguished
The candidate provides no outside resources from professional mathematics organizations.	The candidate collects outside resources but they are not related to the mathematics of the unit.	Appropriate resources are collected and the candidate explains how they relate to the content of his or her math unit.	The outside resources selected by the candidate either effectively extend students’ learning of a topic in the unit or provide scaffolding to students in need of support.

In the next post, we will look at one Teacher Assistant’s Outside Resources entry and I will discuss how certain expectations on my part interfered with my ability to accurately evaluate this work.

Are you smarter than a preservice middle school math teacher?

I gave my final for Teaching and Learning Middle Grades Mathematics on Wednesday; it had three parts. Learners completed the Teaching Math Workshop before class. The Learning Math Workshop was a collaborative effort completed during the first half of the time allotted for the final. During the last hour, learners worked individually on the Doing Math Workshop portion of the final.

The course is designed so the learners gradually take greater responsibility for their own learning. I find the workshop structure very supportive in attaining this goal. Consequently, the final Doing Math Workshop is little more than a basic set of instructions, followed by stripped down items from our text, and concluding with a self-evaluation. Check it out for yourself:

Doing Math Workshop (Final Exam)

TLW demonstrate his or her ability to model whit it means to do math.

Needs: 60 minutes, this workshop, and NCTM Process Standards (optional)

• You can begin by reviewing NCTM Process Standards for Problem Solving, Representations, and Reasoning & Proof. [schema activation]
• Next you need to read through the items provided and identify which will best support you in demonstrating what it means to do mathematics. [focus]
• You can work on one, two, or three items to showcase your ability to problem solve, use representations, and reason mathematically. [activity]
• Finally, you will need to go through your work and make explicit places where you are using the process standards. [reflection]

Problem Set

1. 0.75 divided by 0.3
2. x/9 = 7/12
3. The Susa tablets from the vicinity of Babylon contain tables with relationships of the area of regular polygons to the square of the length of the side.
4. The Type 9 pentagons discovered in February 1976 have four of the five sides congruent and angles measuring A, B, C, D, and E with the relationship 2E + B = 2D + C = 360. (The non-congruent side is between angles with measures D and E.)
5. Compose two reflections over intersecting lines.
6. In one form the Pythagorean Theorem states that if a right triangle has legs a and b and hypotenuse of length c, then squares may be constructed on the sides of the right triangle and a² + b² = c². What happens if … ?

Self-evaluate your communication of your thinking using our 4 Cs + 1

C	Self-evaluation and evidence
Clear What you did
Complete How you did it
Coherent Why you did it
Consolidated How it relates
Connected Where it leads

Is direct instruction a better approach to teaching math?

I received the following in my email today:

An intriguing question that I often wonder about myself. Cambourne's research found that learning requires engagement, which means it depends on how engaging the lecture is to the learner. I clicked on the link hoping for some clarity. I was disappointed.

Here is a sampling from Paul E. Peterson’s article, Eighth-Grade Students Learn More Through Direct Instruction, reviewing the research:

As an instructor myself, I’ve had trouble making up my mind. I can cover a lot of ground in classes where lectures consume about two-thirds of the time. But those classes get less enthusiastic student evaluations than some smaller classes where students are encouraged to solve problems through discussion. I, too, like those problem-solving classes. They require less preparation and are easier to teach.

Before we more on, here are my reactions to this portion of the article. The first is nit-picky, but when Peterson says, “I can cover a lot of ground” it is a red flag for me. I, too, cover more material through lecture, but research shows that many students fail to cover the same ground or retain any memory of the landscape. Second, there is the statement, “I, too, like those problem-solving classes. They require less preparation and are easier to teach.” All I can say is that if planning a problem-solving lesson requires less preparation, then it is not really a problem-solving lesson.

Peterson continues:

So when Guido Schwerdt and Amelie Wuppermann of the University of Munich figured out a way to test empirically the relative value of the two teaching styles (see “Sage on the Stage,” research), it is worth trumpeting the findings. These analysts took advantage of the fact that the 2003 Trends in International Mathematics and Science Survey (TIMMS) not only tested a nationally representative sample of U.S. 8th graders in math and science, but also asked their teachers what percentage of class time was taken up by students “listening to lecture-style presentations” rather than either “working on problems with the teacher’s guidance” or “working on problems without guidance.” Teachers reported that they spent twice as much time on problem-solving activities as on direct instruction. In other words, U.S. middle-school teachers have drunk deep from the progressive pedagogical well.

It is important to note that the results are based on teachers' reporting instructional approaches and not direct observation. This is important because Stigler and Hiebert (1999) found that: “Although most U.S. teachers report trying to improve their teaching with current reform recommendations in mind, the videos show little evidence that change is occurring. Furthermore, when teachers do change their practice, it is often in only superficial ways.” (The Teaching Gap p. 12) This also comes from TIMSS data. However, there is no corroborating evidence in Schwerdt and Wuppermann’s study that supports teachers’ claims that they are using problem-solving approaches.

Furthermore, as the Learning Pyramid shows, not all direct-instruction methods are equal in their effectiveness. A demonstration (think aloud) would be a superior method to simply “covering content” through a traditional lecture. Again, Cambourne has shown that demonstrations are a necessary condition for learning but not sufficient. Learners must be given the opportunity to take responsibility for their learning and “give it a go.”

None of this seems to matter to Peterson who ends his review with, “Sadly, U.S. middle-school pedagogy is weighted heavily toward problem-solving.” In my opinion, what’s sad is that he would try to pass this research off as settling what is clearly a complex issue.

My reading of the Schwerdt and Wuppermann study suggests that it tries to answer the question, “Is traditional teaching really all that bad?” without considering how their methodology might misinterpret the data. I already discussed the problem with associating teacher reporting with using observable data. I am also concerned that they combined, “working on problems with the teacher’s guidance and working on problems without guidance” into the single problem-solving category.

Watch this lesson of a U.S. classroom where students are "working on problems with the teacher’s guidance" (from the original 1995 TIMSS research) and decide for yourself whether this teacher “drunk deep from the progressive pedagogical well.” (You will need to sign up for a password but it is free.) My answer is, “No,” but this would be categorized as problem-solving time in the Schwerdt and Wuppermann study.

“Is traditional teaching really all that bad?” – based on this research, the jury is still out.

How will you organize your classroom of the future?

This past Wednesday was the last day for my Teaching and Learning Middle Grade Mathematics course. It was also the end of our unit on teaching mathematics, which meant developing an anchor chart to represent our current understanding (much like we did for doing math and learning math). Instead of having them summarize the content, I asked them to consider what their future classroom might look like given what we knew about doing, learning, and teaching mathematics.

The following video sums up the activity. This group of preservice teachers considered the role communication (learners placed in groups) and technology would play in their classroom but then they stopped and thought about their future learners. The result was surprising to me (the reason I like open-ended activities like this).

Cynically, I might think that they were trying to avoid the work, but they had already done the work. The final vision of their future classroom represented an authentic concern about meeting learners where they are at. The other groups agreed and I felt good about what they had learned this semester.

We spent the rest of the class watching a TED Talk and discussing the future of education. It wasn't the end-of-the-year party that a lot of classes were having, but it was a celebration. I love my job!