Where should we focus our efforts?

During the Second World War, Allied bombers were sustaining heavy damage from flak and bullets while flying missions over Germany. Many of these bombers failed to return to their bases in Britain. A study was conducted of the damage done to the planes that did make it back which resulted in a proposal to add armor to those areas that had sustained the greatest number of hits. This would have been a costly undertaking as it meant spending valuable resources on refitting the planes. Furthermore, the extra armor would make the bombers heavier and less maneuverable. Had this effort been carried out, it most likely would have failed.

From Digital Roam

Fortunately, a Hungarian mathematician, Abraham Wald, took another perspective. Why not reinforce those parts of the bomber that had not been hit? He had noticed a different pattern in the shot-up planes. Each had avoided damage in similar spots, and Wald reasoned that these were vital areas that needed to be protected. After all, these bombers had successfully returned from their missions in spite of the damage they had received.

The Allied Command followed Wald's suggestions. (A collection of his memos can be found here.) However, I believe some loud and persistent voices in education reform are missing the wisdom of his message. Recent videos developed by Students First exemplify this misunderstanding of what is vital to the success of the U.S. education system. (Students First is the group headed by former Chancellor of the Washington D.C. school system, Michelle Rhee.)

The video shown here suffers from the same problem that distracted the original researchers of the WWII British bombers. Students First is focusing on the testing data (where we were "hit") and ignoring what is really important in education. To continue the Olympic metaphor, it would be like judging our divers against those of other nations simply based on how long it takes to climb the ladder. It does not matter, which makes it a distraction, and a potentially costly one.

Dr. Yong Zhao explains the problem with this focus on using only tests to compare our educational system to other country's in his keynote at the AACTE 2012 Annual Meeting. The keynote starts about 18 minutes into the video.

Education reform groups like Students First say that our current standardized test results show that our education system is full of holes. The truth is that by that reasoning, "American education has always been bad." [32:33] Poor test scores are not a recent phenomenon. "We have had over a half-a-century of bad education according to some measures." [40:49]

Much like Wald did, Zhao interprets these "hits" differently. In the figure below, the countries on the left are the top ten countries in the Programme for International Sudent Assessment (PISA). Those on the right represent the leaders from the Global Entrepreneurship Monitor (GEM). 

"As you can see the two lists don't go together. Countries that have higher PISA scores have lower entrepreneur capabilities." [48:40] In other words, focusing on high test scores means embracing conformity at the expense of what has always been our national strength - creativity.

Unfortunately, the current wave of education reform focuses on the easy to observe gaps rather than our subtle and vital successes. Yes, we have holes but these holes have not kept us from being successful. Still, we seem committed to embracing standards-based test-prep while eliminating the aspects of our educational system that nurture creative thinking. If we do not listen to the wisdom of Wald and Zhao, we will spend a lot of resources to fix what is not really important and make our education system less flexible as a result.

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.

When will we ever use this?

A couple of weeks ago, this comic showed up on xkcd.

Forgot Algebra

As a math teacher, I can relate to the sentiment of this comic. People regularly boast to me about how bad they are at math (especially algebra), and still, they are successful in life. Much like the young woman in the comic, these people seem to believe that they were lied to about the utility of math, and they are resentful about it. Maybe lied is too strong a word, but perhaps we do misrepresent the purposes behind the math taught in schools.

Because I was teaching some algebra to middle school students twenty years ago, I remember how I responded to the question, "When will I ever use this?" Therefore, I have a theory about what might have been going on in Miss Lenhart's Algebra classes in the late 80s/early 90s. I imagine it might have looked like one of these four scenarios (depending on what she tried before).

Has much changed? I mean, besides the fact that we now add cell phone contract problems into the mix. For the most part, students remain unimpressed as we tie ourselves in knots to demonstrate how they will need the math we teach in everyday life. That is why I suggest that when asked by students, "When will we ever use this?" that teachers respond truthfully.

I don't know when or if you will ever need this particular concept. It depends on what you do with your life and what technological advances are made in the future. But you know what you will need to be able to do, regardless? You will need to problem solve. You will need to think critically (reason and prove). You will need to be able to communicate quantitative thinking to others. You will need to use representations to support your thinking and share your thinking. And you will need to make connections in order to consolidate your understanding. Mathematics is a discipline that provides opportunities to practice and strengthen all of these skills. So, as we solve for x, I want you to monitor your thinking because that's what's really important.

That is how I learned to respond to my middle school math students. The NCTM Process Standards gave purpose to my math lessons, and the students bought into it. It worked for me because I finally believed in what I was teaching.

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.

Do you have any Yis (Ones)?

Having successfully decoded the Wumanian number system (and further developed their understanding of place value), the preservice teachers begin to explore the concepts of composing and decomposing numbers. This is done through the use of centers - an instructional approach used successfully in many elementary classrooms. In this post, I share some of the more popular activities.

Yiwu Go Fish

As with many of the Wumanian center activities, this game has an Earth equivalent.

Summing Yiwu-yi Cubes

• With a partner, take turns rolling the dot cubes;
• Using symbols, write the amount shown on each individual cube (see example below);
• Write the total of the cubes combined (see example below); and
• Pass the cubes to your partner and repeat.

Collecting Coins

• Use a pencil and a paper clip to create a spinner out of the sectioned circle provided below;
• Take turns using the spinner to determine how many coins to add to your collection;
• Each turn, say how much you had, what you spun, and your new total after adding your spin.
• Keep track of the total worth of the collected coins using the rekenrek;
• Keep spinning until you reach a total of erwuwu pents; and
• If you finish early, play the game in reverse - giving the amount spun away until you reach na pents.

Fact Finding

On the addition table, identify which facts fit into which categories (note that some facts will need more than one color):

• Doubles
• Doubles plus or minus yi
• Count up yi
• Sums of yiwu
• Bounce off yiwu

Story Problems

In this center, learners work to solve a selection of Cognitively Guided Instruction "Joining" stories.

• There are si people in this group and si people in another group. How many rekenreks will I need if every person gets a rekenrek?
• I have er dot cubes. This center requires si dot cubes. How many more dot cubes do I need?
• A puzzle is made up of yiwu-yi pieces. Some of the pieces are already put together. There are san pieces left to complete the puzzle. How many pieces are already put together?

Yiwu Fingers

Yiwu is a benchmark number in the Wumanian system. Students who learn the different ways to decompose this benchmark are often more successful learning their facts and later on developing efficient mental strategies for larger numbers. This song and the accompanying motions (putting up the appropriate number of fingers and spinning around at "Now it's time for turn-arounds") is one way to support students in this area.

***

After the preservice teachers rotate through the centers, they are asked to reflect on the experience. I have them focus on how they might apply what they did in each activity to teaching composing and decomposing numbers in our system. Their responses form an assessment that allows me to evaluate whether or not they are able to transfer their experiences in base-yiwu to our base-tem system. Then I can then plan future learning activities based on their current level of understanding, not just what comes next in the curriculum.