# Mental Math

My sister gets nervous sometimes when we are at the store together.   She is afraid (rightfully so) that she may have to witness me give a math lesson at the cash register.   I can see why it might be embarrassing to her and I can also see why it infuriates people behind me in line, but sometimes I just can’t help myself.     As a math teacher, it frustrates me that the cashier can’t figure out the change when I give her a twenty, a one and a nickel for a bill that is \$11.05.

Now, one could argue that the reason why the cashier can’t do this is because he or she did not receive enough instruction on the traditional algorithm for subtraction.  Maybe he or she used calculators too much in class instead of doing lots of “drill and kill” exercises.

But I don’t think that having lots and lots of practice with the traditional subtraction algorithm would help in this case.

The truth is, I don’t often use traditional algorithms when solving math problems in real life.  I do use a lot of mental math.

I count up or I count down when adding or subtracting.  I break numbers apart when multiplying.  I rarely think about multiplying 25 times 6 in my head using a traditional algorithm. Instead, I think of 20 x 6 plus 5 x 6 or I think about 25 x 4 plus 25 x 2.  It is just easier.  To figure out a 15% tip, I usually find 10% and then half it and add the two together.   Or most of the time I double it because waiters and waitresses really deserve that 20%.

I bet a lot of you operate the same way.  You don’t take out a pen and paper in the store or restaurant.

Yet as I mentioned in a previous post, some people still think that traditional algorithms are the only way to teach math.

As you may expect, I disagree.  How much more valuable is it to work flexibly with numbers? To be able to calculate things in your head quickly.  To have a sense of whether an answer is way off base.  I think this is the way we should approach math in the early grades.  It certainly would make my experiences at the grocery store less traumatic.

So what does the research say? Two researchers (Fosnot and Dolk, 2001) reported on a program called “Mathematics in the City” in which teachers fostered mental math computation strategies in children between the ages of 4 and 8.  They showed how these students developed a deep understanding of number and the operations of addition and subtraction.

Now is there a place for learning traditional algorithms?  I think there is.  But I think it’s often more valuable to do after students have had experience working flexibly with numbers.

So how do we begin to do this in our classrooms?    That I’ll talk about in my next post.  But feel free to write any suggestions you have in the comments below.

Want to know more?

Check out the Mathematics in the City site here.

Check out Fosnot and Dolk’s book: Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction.

# Dan Meyer’s Makeover Mondays: Modifying tasks

As I mentioned in a previous post, not all tasks are created equal.    Different tasks provide different opportunities for students to learn.

Deciding what task to use in a lesson or how to modify an existing task is a decision we make every day as teachers.

Dan Meyer recently highlighted how we can begin to think about modifying tasks with his Makeover Mondays.  Every Thursday this summer, he will post a task on twitter.  The following Monday, he will post his suggested modifications as well as the changes others have suggested.

What I love about this is that we know from the research that teachers are constantly changing tasks.  Tasks are not always implemented exactly as they appear in textbooks.    Sometimes the task doesn’t fit the needs of our students or it doesn’t build on their prior knowledge or sometimes it is just boring.

However, we also know that sometimes when teachers change the task, it can lower the cognitive demand of it.   Researchers have found that when high cognitive demand tasks are implemented in classrooms, the demands were often lowered because of things like time constraints or management issues in the class.    A mathematically complex task became easier as the teacher gave hints or leading questions to help the students proceed.

But what is obvious from the makeovers Dan posted is that the changes that teachers make are far more complicated than simply lowering or increasing the demands of the tasks.

The made-over task Dan posted this week was more cognitively challenging than the original task, but also built on students’ intuition.  The language was simplified but the mathematics was increased so that students who completed the task easily were challenged.

I’d also like to point out another thing I noticed about modifying tasks–the goal matters.  If the goal is application of a concept, the task needs to be modified differently than if the goal is to introduce students to a concept.  If the goal is to have students model different strategies, the task needs to be modified so that it lends itself to multiple ways to be solved.  If the goal is to work on problem solving and not computation, perhaps a calculator needs to be provided.

Tasks can be modified to serve different purposes.  But it’s important to be clear on what your goal for your students is before you think about changing it.

So this summer, take a look at the tasks.  Share a makeover.  I know I’m looking forward to trying my hand at it.

# The “Math Wars” still rage on

What amazes me about working in schools is that when I walk into one I am immediately transported back to being a student.  The clothes may be different and the hairstyles may have changed, but in the over twenty-five years that has passed since I was a student in an elementary school the model for teaching math still looks very similar in a lot of classrooms.

The teacher stands at the board (maybe now it’s a smart board), models an algorithm and the students practice it at their desks.  Now I’m not saying that this happens in every school.  I’ve had the chance to visit classrooms that use very different approaches and I like to think that when I was a teacher, I pushed for an inquiry-based way of teaching as much as I was able to at the time.

But it’s perplexing to me that 25 years of research haven’t been able to change the traditional model in a lot of schools.  Imagine if the hospital I went to 25 years ago still used the same methods today.

However, after reading the recent op-ed piece in the New York Times that argues against math education reform, I see why it is so hard to enact change.   There are still many people who don’t buy into new methods of teaching math, hence the so called “math wars.”   The fact that this debate between traditional and reform methods of teaching still wages on is shocking to me because we know that our traditional ways of teaching are failing lots of students.

After reading the piece in the Times, you should check out Professor Keith Devlin’s thoughtful paragraph- by-paragraph critique of the article.   It makes a number of good points that I won’t rehash here.

The thing that bothered me most about the op-ed was that there was no connection to what we know about how students learn.  There is no attention given to what the research says about student learning.   There has been plenty of work (see the references at the end for some of it) that demonstrates that students learning math in reform-based classrooms outperform students in traditional classes and these students report stronger motivation and interest in math.  What’s more, the reform approach has been shown to be successful with students of diverse backgrounds.

It just doesn’t make sense to me to keep doing what we’ve been doing and expecting different results.

What do you think?

Want to know more? Check out some of the research below.

Boaler, J. (2002). Experiencing school mathematics: Traditional and reform approaches to teaching and their impact on student thinking. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

Knapp, M. S., Adelman, N. E., Marder, C., McCollum, H., Needels, M. C., Padillia, C., Shields, P. M., Turnbull, B. J., & Zucker, A. A. (1995). Teaching for meaning in high poverty schools. New York: Teachers’ College Press.

Silver, E. A., & Stein, M. K. (1996). The QUASAR project: The revolution of the possible in mathematical instructional reform in urban middle schools. Urban Education, 30, 476–521.

Van Haneghan, J. P., Pruet, S. A., & Bamberger, H. J. (2004). Mathematics reform in a minority community: Student outcomes. Journal of Education for Students Placed at Risk, 9(2), 189–211.

# What can we learn from Finland?

Whenever I tell people I research teaching and learning, they inevitably ask me about Finland.

In the 1980’s, Finland transformed their education system.   Student achievement scores increased.  Teaching became a prestigious career choice among young people.  Teacher preparation programs became one of the most competitive systems in the world.   As a result of these changes (and more), the country comes up often as a promising model for educational reform.

When people talk to me about Finland, the conversation usually turns one of two ways.  In the first, they tell me that Finland is too different from the United States for us to learn anything from them.   They argue that the country is much smaller and not as culturally diverse and that it would be impossible to scale up what is being done in Finland in the U.S.  While there are certainly limitations about what we can learn from them, I’m not sure it’s in our best interest to, as the saying goes, throw the baby out with the bathwater.

The second group of people often ask me why we don’t just copy whatever they are using in schools Finland and use it in the current system here.   They argue we should use their curriculum materials or we should hire teachers that are trained there.

It’s an interesting suggestion, but what would happen if we did that?  Pasi Sahlberg, a researcher and writer on school reform, wrote about exactly that in this piece for the Washington Post.

He argued that if Finland’s teachers were placed in schools in the U.S., they wouldn’t be able to produce any significant gains in student achievement.  The policies and systems in the U.S. would limit the Finish teachers’ abilities to produce the gains they were able to produce in Finland for a variety of reasons, including the pressures of standardized tests, inflexible curricula, lack of strong leadership from principals and lack of support from homes.

Now, this doesn’t mean there isn’t anything to learn from Finland, just that we need to be careful about asking the right questions and how we apply what we learn.  I think that’s where research comes in.

I’ll discuss more about the research in future posts.  But if you want to learn more right now, check out Pasi Sahlberg’s book Finnish Lessons.  It’s talks about the research in an engaging way and provokes some interesting questions about education reform.

# Do career and technical high schools improve math skills?

Last week I had the opportunity to tag along as the Secretary of Education, Arne Duncan, visited Aviation High School in Long Island City.  He was there to highlight the school as a model for preparing students to become both career and college ready.  You can read about more about the visit in my post on the Department of Education’s blog.

What struck me about the school (besides how neat it was to see the planes up close) was how engaged and motivated the students were as they talked about repairing planes and interning for businesses at nearby airports.

The students talked about how their aviation classes helped them perform better in traditional classes like physics and math, which made perfect sense to me.  In the case of this particular high school, it certainly was the case.

However, I wanted to see what the research had to say, in general, about career and technical education classes and their impact on math achievement.

As is often the case, the research paints a more complicated picture.   Many high school students, including ones who are enrolled in career and technical courses, do not graduate with the math skills necessary for the workplace or for college.

A group of researches (Stone, Alfred & Pearson, 2008) argued that while career and technical courses hold potential for improving math skills, the math that is being used in these courses is not always being made explicit.

They set out to test what would happen if teachers of technical classes (who are not usually trained as math teachers) made the math explicit.  For example, what if the shop teacher talked explicitly about the Pythagorean theorem during a lesson on the T-square in carpentry class?

They found that when technical and math teachers worked together to make the math explicit in these courses, students performed better on traditional and college placement math tests.  In addition, this type of intervention did not negatively affect their content knowledge in their occupational areas.

If I had to guess what was going on in Aviation High School, my guess would be that it is something similar to what the researchers proposed in their intervention.  But I think it’s important to note that while engaging students in hands-on activities and real world situations is part of the story, it’s not the whole story.  How to best connect mathematics and real world learning needs to be thoughtfully considered.

Want to know more?

You can read more about what the students had to say during Secretary Duncan’s visit in my post on Homeroom, the Department of Education’s official blog.

Read the research:  Stone, J. R., Alfeld, C., & Pearson, D. (2008). Rigor and relevance: Enhancing high school students’ math skills through career and technical education. American Educational Research Journal, 45(3), 767-795.

# What matters when choosing tasks for students?

“Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.”  -Stein, Smith, Henningsen, & Silver, 2000

What are we doing today?

One of the biggest decisions we make as teachers is choosing the tasks our students will work on during class.  We use tasks from textbooks and the Internet.  We borrow tasks from colleagues or design our own.   No matter where the tasks originate from, the nature of tasks we select affects our students’ ideas about what it means to do mathematics.  Different tasks provide different opportunities for students to learn mathematics.

One way to think about how to select a task is by looking at the cognitive demand required by it.   Cognitive demand refers to the type of student thinking required to solve the task.   A group of researchers developed different categories of cognitive demands found in mathematical tasks.

Low-level cognitive demand tasks can be solved by memorization or by using procedures that don’t have meaning for students.  High-level tasks engage students in using procedures with attention to the reasoning behind them or in what they label “doing mathematics”–making conjectures, justifying, and interpreting.

After studying over 500 tasks in middle schools, the researchers found that the greatest gains in student achievement occurred in classrooms where teachers used high-level tasks and the cognitive demand of the task was maintained as students worked on the tasks.   It was also noted that although many tasks started out requiring high cognitive demands, the demands of the tasks often decreased as they were implemented in classrooms.

Of course, there is a lot more to think about when selecting tasks.  How do the tasks build on what your students already know?  How do the tasks engage students?   What particular mathematical concepts are fostered by the task?  But considering the level of thinking you want to foster in your students is a good place to start.

Want to know more?   Check out the research below.

Stein, M. K., & Lane, S.  (1996).  Instructional tasks and the development of student capacity to think and reason:  An analysis of the relationship between teaching and learning in a reform mathematics project.  Educational Research and Evaluation, 2(1), 50 – 80.

Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. Teachers College Press, 1234 Amsterdam Avenue, New York, NY 10027.

# Another Tool for Implementing the Common Core

It’s hard to have a conversation with anyone in education these days without the two C’s coming up: Common Core.  Now I know there are lots of opinions on whether we should abandon them, whether they create an increased focus on testing, and whether they are suitable for all students.  I’m not going to weigh in on any of those debates.  At least not right now.

What I am going to do is to talk about is a resource that might help you if you are trying to implement common core standards in your classroom.   In an earlier post, I talked about one resource I find particularly helpful: http://www.turnonccmath.net/.  I received a lot of positive feedback about the tool from both teachers and parents.

So when I stumbled across another resource that I think is useful, I thought I would share it.  The website Illustrative Mathematics  provides tasks, videos, lesson plans, and curriculum modules that illustrate the Common Core Standards.  What I like about the site is that the tasks are submitted by a variety of individuals:  teachers, researchers and mathematicians.  Before being posted, each task is reviewed by both a classroom and mathematics expert.  I think this collaboration across different groups allows for a rich set of tasks.

It’s still a work in progress so not all standards have accompanying tasks or lessons plans.  However, I’ve found the K-8 tasks that are posted to be very helpful.  What I really like is that you can easily download and print a task for a particular standard you are teaching.  Included with the task and solution is a commentary on the purpose of the task and some suggestions on modifying or extending the task.

How you use these tasks with your students is up to you.  Some are better suited to be used as assessment items, while others are well suited for whole class lessons.   Your professional judgment, as well as your understanding of your students, will help you to decide how to best use the tasks.

I’d love to hear if you find the site useful.

# How do you know? Part II

In a previous post, I talked about why it might be important to ask a student “How do you know?” when they give an answer to a problem.  What I neglected to talk about was the research that supports this recommendation.

Luckily, Education Week recently published an article about several research studies that claim asking students for explanations can deepen their understanding.

Although the research mentioned does not talk specifically about learning math, it does lend support to the idea that asking students for explanations is important.  Furthermore, one study suggests that asking for an explanation focuses a student’s attention on what the underlying concept is as opposed to other aspects of the task.

What I did not see addressed in the research presented was how we can better foster explanations in our students.   It cannot be enough to simply ask them for explanations or to focus their attention on figuring out how something works.  It seems to me that we need carefully designed tasks that allow them to develop an understanding of why something works so that they can explain it.

Asking for an explanation is a good suggestion, but what do we do when students can’t explain?  I’d love to hear your thoughts.

Want to know more?

Access the article (which includes links to the accompanying research papers) here.