Where’s the Math

After doing my taxes this past weekend, I realized that I did so without doing any math. I just put numbers into various boxes and trusted the software to do the rest. Perhaps the only math involved was having a sense whether those numbers I was entering seemed reasonable. This made me start to wonder about the math most adults do in their daily lives. How many people use the quadratic formula in their daily lives? Yet, when they learned it, did they learn it in a valuable enough way, that with that new knowledge, they can think in a particular way? How many know that when there are six people dining and you split the bill evenly, leaving a 20% tip, all you have to do is just divide the bill by five and have the sixth person cover the tip? If your student is working on 3-digit by 3-digit subtraction and on a post-test makes many errors, can you tell what directly caused those errors?

I ask that last question because as a school we’ve been examining several math curricula. One of them has an incredible technology component that includes computer based assessments. It’s amazing how quickly you get data back and the teacher doesn’t even have to grade the paper. Easy, right? Upon further reflection though, a child who might still get about half the questions directly involving 3-digit by 3-digit subtraction wrong, the data would simply just indicate that. Without examining the scratch piece of paper, interviewing your student, or observing the child in action, you wouldn’t be able to isolate whether or not the error was a simple fact error, errors with regrouping, inversion, or even adding instead of subtracting. If you were able to isolate what that error was, though, imagine how quickly you could help that child develop.

This month’s issue of the journal, Teaching Children Mathematics, contains a few great articles. One is called, “Action Research Improves Math Instruction,” which features elementary school teachers who, as part of a course they’re taking, embark on a “practitioner-based” research process in their classrooms. One of them, a 3rd grade teacher, looked carefully at 3-digit subtraction, read about the kinds of common errors children make on questions like these and decided to make her students ‘subtraction detectives.’ They had equations that were already solved, some with errors, and they had to practice finding and describing the error. The improvement in her students’ assessments improved greatly. The teacher didn’t know whether this was a ‘best-practice’ but it made solid sense to her and she gave it a try. The article mentions that “Action research addresses specific student needs, targets classroom issues, keeps teachers current, and discourages ineffectual methods.”

This year, our school has been examining several different math curricula with one of its objectives being a common scope and sequence. Today, we had a faculty meeting discussing the pros and cons of the different curricula, and I found the discussion rich and robust. We also asked ourselves some very important questions. We didn’t come up with any immediate answers, but I was really impressed when colleagues disagreed with each other, how the discourse remained passionate, but civil, and everyone made extremely insightful and thoughtful comments. Everyone seemed to be aware of their own biases as they spoke. I wondered, leaving that meeting though, and re-reading this article, if we needed not only to think of a common set of expectations, but if we could also find ways to examine student progress even more carefully and identify where gaps lie, or how their learning can be enriched.

Another article in the same issue called, “Professional Development Delivered Right to Your Door.” It listed the following as Best Practices of Professional Development: Professional Development must be -

  1. grounded in participant-driven inquiry, reflection, and experimentation;
  2. collaborative, involving a sharing of knowledge among educators and a focus on teachers’ communities of practice rather than on individual teachers;
  3. connected to and derived from teachers’ work with their students;
  4. sustained, ongoing, intensive, and supported by modeling, coaching, and the collective solving of specific problems of practice;
  5. related to other aspects of school change; and
  6. engaging, involving teachers in concrete tasks of teaching, assessment, observation, and reflections that illuminate the processes of learning and development (Darling-Hammond and McLaughlin 1995).

Regardless what direction we go in math, I feel like we met all those goals. I think the process was, and will continue to be an ongoing one. I feel very fortunate to work at a school with such caring and passionate teachers.


Knowing What AND How to Teach Math (then reflecting on it so you can do it even better next time)

The November issue of Teaching Children Mathematics features 3 very good articles. The first, “What Knowledge does teaching require?” by Thames and Ball mention that a clear description of such knowledge in research over the past 40 years has been rather elusive. Just because you know a lot of math or can do math very well, doesn’t mean you can teach it well. Of course, the authors do not doubt that you have to have a good grasp of mathematics. Whether you’re a math major, however, does not make a difference according to their findings. What they argue is that teachers need to “uncover the mathematical issues that arise in practice. By better understanding the mathematical questions and situations with with teachers must deal, we would gain better understanding of the mathematics it takes to teach.”

They broke up what they called, “Mathematical knowledge for teaching” into several domains in relation to teaching. There’s “subject matter knowledge (SMK)” and “pedagogical content knowledge (PCK).” Each of these two can be divided further into three sub categories each. SMK can be grouped into these three: common content knowldedge, knowledge at the mathematical horizon, and specialized content knowledge. PMK can be grouped into these three: knowledge of content and students, knowledge of content and teaching, knowledge of curriculum.

The authors also mention that these are some math skills good teachers or mathematics need:

  • posing mathematical questions
  • giving and appraising explanations
  • choosing or designing tasks
  • using and choosing representations
  • recording mathematical work work on the board
  • selecting and sequencing examples
  • analyzing student errors
  • appraising students’ unconventional ideas
  • mediating a discussion
  • attending to and using math language
  • defining terms mathematically and accessibly
  • choosing or using math notation

Just today, we introduced the children in my class to base 4. The purpose is to solidify the concept of place value and to understand the construction and deconstruction of numbers so that the children have a better understanding when taught the traditional algorithm. Just going through the list above, we did several things differently from last year. This year, I chose to create something that we could represent on the board. It made a huge difference – not only in understanding, but in engagement. Of course having kids interact with the board was great. So what if the architects decided that the board had to be a minimum of so many feet off the ground, I stuck a chair nearby and those who needed it, could step on and still reach. Letting the children give their explanations as to why they were rounding up groups of 4 and putting them in the next column seemed to help a lot too. Posing questions was also great. About halfway through the activity, one of my students raised her hand and said, “I think that with the materials we have here, the highest number will be 133.” She was absolutely right.

This article says that mathematical knowledge in the way that is required for reasearch in mathematics is not what is needed. What is needed is someone who has excellent “mathematical knowledge for teaching which itself is a kind of complex mathematical understanding, skill, and fluency used int he work of helping others learn mathematics.”

The second article, titled “Balancing Act” focused on high-quality professional staff-development. First of all, that development needs to be ongoing. According to the author, it needs to be focused on three essential things:

  1. Content focused on instructional design and implementation
  2. Process characterized by inquiry, observation, research, and collegial interactions
  3. Contextual support for job embedded professional training

This development has to be long-term and sustained in order to make a lasting change. If you think about those three criteria above combined with the need to customize learning to the needs of each individual student, there is no way that two classes will ever be the same, and while a textbook may be a useful tool at times, one must resort to using other tools as well. The author of this article says the challenge is to find the time to collaborate and reflect on what worked and what didn’t. The author goes on to say that textbook providers tend to provide training to teachers once they’ve purchased their materials, but that training goes away after a year or two and teachers, especially new teachers, need continual training. Another challenge that lies in changing how math is taught is the resistance many teachers have to changing what they’ve been doing for years. I have always been open to change, but I have to admit that I am resistant to thinking that a new textbook alone can be a panacea to improve the teaching of mathematics. You need to also know HOW to teach it.

Finally, the third article from this issue I liked was titled, “A reflection framework for teaching math” by Merrit, Rimm-Kaufman, Berry, Walkowiak, and McCracken. While these authors offer a guideline with 8 dimensions in their “framework for reflection, not every lesson will be ‘high’ on all dimensions.” Below are their 8 dimensions. As one reflects on a lesson, here is what to consider:

  1. Lesson structure
  2. Multiple representations
  3. Mathematical tools
  4. Cognitive depth
  5. Mathematical discourse community
  6. Explanation and justification
  7. Problem solving
  8. Connections and applications

After we finish working in different bases in my class, we will move on to addition and the use of traditional algorithms. There are some children, solid in their understanding of place value, who already know how to use the traditional addition algorithm where you carry (regroup) into the next column. If these children have already mastered this, how do I extend their learning while teaching the rest of the children the traditional base-10 algorithm for addition? At least I’ll have until after Thanksgiving break to ponder this.

My opinion on why so many districts have failed to implement a textbook series successfully is not because of the content within those books (as I mentioned they make a nice tool – and there are many online tools available for free, if one is willing to take the time to find them), it is because there hasn’t been sustained development in making better teachers of mathematics regardless of the text or tool. Whether it be Everyday Mathematics, Singapore Math, or other programs (each has it’s pros and cons), it boils down to how you use it with your students. The concepts behind the math may not have changed over the years, but the way we teach it, engage our kids, and make it stick certainly have.


Again, in Math – It’s About the Process

Later this week I get to sit down with parents and together set some concrete goals for their children. It’s a great process as it allows us to focus and target as well as individualize what we think will be most beneficial to a particular child. To increase success and motivation, I would even include the child in setting their own goals.

Our faculty also sit down with our administrators and have a goal setting meeting and I only had one goal this year – to get better at what I do.  I then broke that one goal out into little goals, and I really don’t know how my one goal morphed into four pages – especially when my first bullet point was to simplify things. Nonetheless, I still only have one goal – to grow as a teacher. Anyway, one of my ‘sub-goals’ if you will, was to read one article from Teachers of Mathematics a month since math continues to be an area of curricular focus for our school this year.

The one I picked from the October issue was called Building Word Problems: What Does it Take? by Angela T. Barlow.

The first question asked by the author is why would we create our own problems when there are text books and other curricular materials out there. The reason is pretty simple. Because those materials (while they have value) were not written for every kid in mind and creating your own problems allows you to customize learning. Furthermore, many of those problems focus on a single strategy vs. having kids use what they already know. The authors also say we need to thisbecause “future citizens must be prepared to problem solve and apply their skills to new situations.” According to the article, there is a process involved that requires careful thinking. Here’s the process:

  1. Identify the mathematics goals.
  2. Decide on a problem context.
  3. Create the problem.
  4. Anticipate students’ solutions.
  5. Implement and reflect on the problem.

Every morning when the kids come in to school, as they are settling in, I have a math problem on the board. Increasingly, I’ve tried to make the questions more open ended or focus on the process involved. I thought I’d look at the process above and reflect on it here.

The goal was to add two two-digit numbers together, use a variety of strategies, have kids explain their thinking, not rely on algorithms that haven’t been taught, and perhaps get children to round numbers.

The context was just for them to be able to communicate their thinking.

The problem (not very creative) was something along the line of “Add 99 and 99 together. Explain your thinking.”

I anticipated many of my second graders to struggle with this one. I had used it much later in the year last year and noticed kids immediately relying on the traditional algorithm they had learned. Not that there’s anything wrong with that (algorithms are great tools, but they don’t always reveal how one thinks about numbers).

With some children I prompted with further questions like, “is 99 close to a number that’s easy to work with?” Many of the children didn’t hesitate, they tried to use what they had to solve the problem.

With that question, one child said, “Oh, 100 plus 100 equals 200 and you just have to subtract 2 to get 198.”

Another child said, “Well you just add 1 to 99 to get 100. Then you add the 99 to get 199, but don’t forget you have to take away the 1 you added, so the answer is 198.”

One child said they knew 90 and 90 was 180 and that 9 and 9 was 18. They couldn’t seem to add 180 and 18 together, so he just deconstructed it and said, “180 plus 10 is 190 and 190 plus 8 is 198.”

It’s not the greatest question, but it’s really nice to see kids grappling with two-digit addition long before we introduce them to the traditional algorithm. Not only is it promoting good number sense, but it also promotes creativity. They are not relying on a single strategy from a text. They are in fact teaching themselves how to solve a problem. They are becoming more resourceful. Upon reflection, the problem could be asked in a better way, but the outcome for most was a positive one.


Singapore Math Adopted in More U.S. Schools – NYTimes.com

Singapore Math Adopted in More U.S. Schools – NYTimes.com.

This article is an interesting one. As we continue to look for ways to improve math education in our school and across the country (this is a national issue), Singapore Math creeps up again and again.

We use manipulatives daily and go from concrete to visual to abstract in almost all of what we do daily. The objective for the teacher is for the student to understand the math objective for the day. You can learn so much from a child by just asking a few questions, and you can differentiate fairly easily in math. I really believe that ambiguity in math, rather than the ‘one right answer’ approach helps kids develop better thinking about our number systems.

It was nice to get to the end of the article and see that it mentions a 2nd grade class using dice and building two and three-digit numbers. We did that all of last week in a variety of ways to help the kids understand place value. But there are other ways too. Using different base systems, for example.

99 may be 9 tens and 9 ones, but it’s also only one less than a hundred, and sometimes that is really more useful. The latter is not often taught and this year, I am trying to get kids to think about numbers in many different ways. Playing with math is what makes it fun. It’s not about getting right and wrong answers – it’s about the process in how to get there.

I’m looking forward to my little experiments this year and adding more ambiguity to some of our math, so that kids become more comfortable with numbers.

Making Math Meaningful for All

I finally got a chance to dig into my copy of Education Leadership and several articles have already grabbed my attention in a good way. I was glad to see a an article about differentiating instruction in math and some easy ways to do it.

If you’re an ASCD member you can get the article here, but I’ll try my best to summarize or feel free to borrow my copy when I’m done. I also saw one floating around in our faculty lounge.

The article is titled, Beyond One Right Answer and criticizes that math is viewed by too many students as something to answer quickly with an answer that is expected by the teacher. First of all, two widely held beliefs need to change:

  1. That all students should work on the same problem at the same time.
  2. That each math problem should have a single answer.

Two core techniques are discussed in the article. They are not new, but every teacher of math should be aware of them: using open questions, and parallel tasks.

Open Questions

The easiest way to create an open question is to provide the answer to the students and have them challenge themselves to make up the question. At first students can be uncomfortable with ambiguity, but they will warm up.

Another strategy for open questions is to ask for similarities and differences. Ask students to give two examples each of how 4 and 9 are similar and different.

Yet another strategy is to allow choice in the data provided before solving the problem.

Finally, ask students to write a sentence. Give certain vocabulary they must use, but have them design the statement.

Parallel Tasks

These are tasks that focus on the same big ideas but have different levels of difficulty. I remember that each year, we do a quilting unit and when asking kids to describe how they figured out how many quilt blocks were used, we received different suggestions from repeated addition to some kid multiplying the width by the length (we don’t teach the formula for the area of a rectangle in second grade, but it’s exciting when kids discover it on their own).

One way to create a parallel task is to let students choose between two problems. The same concepts could be covered, but one of the questions would be more difficult than the other.

A second strategy is to pose a common question like “what operation would you use to solve that problem?” or “Can you solve this in your head?”

Math is not something you train kids to do like a pet. It’s something that should be explored, discovered, played with, and made real.