# Is a careless error really careless?

In a 1982 study by John Clement, 150 freshman engineering students were given the following problem:

Write an equation using the variables S and P to represent the following statement:

“There are six times as many students as professors at this university.”

Use S for the number of students and P for the number of professors.

37% of the students answered incorrectly and wrote: 6S=P.

Why did they make that error?   Take some time and see if you can figure out why.

At first, Clement thought the mistake was due to carelessness. These were engineering students and this was a simple algebra problem.  They must have just been careless, right?

Clement wanted to check his hypothesis.  Using clinical interviews, he asked students to think aloud as they solved the problem.  At first, he thought he was correct in assuming it was a careless error– it appeared that students were using what he called a “word-order-matching approach.”  They incorrectly mapped the order of the key words in the problem directly to the order of the symbols in the equation.

However, as he continued his analysis of the interviews, he realized there was another explanation.   Students who incorrectly answered the questions were doing something that made sense to them.   Their intuition was to place the multiplier next to the letter associated with the larger group.   Although incorrect, it was meaningful to students.

The correct answer, S=6P, was not meaningful to them.  It did not describe the situation in the problem.  Professors are not multiplying.  It did not make sense.

Yet it makes sense to us and to the students who could answer it correctly.   Clement explains that we understand it because of 2 reasons:

1. We see the variables as representing numbers rather than objects

2. We are able to invent a hypothetical operation on the variables that creates an equivalence.  We know the equation represents what would happen if you made the group of professors six times larger than it really is.  Then the number of professors would be equal to the number of students.

The problem doesn’t seem so simple now, does it?   If we attributed students’ mistakes on it to a careless error, we wouldn’t diagnose the problem correctly and if we don’t diagnose the problem correctly, we have little chance of figuring out how to remediate it.

This happens a lot in classrooms.  We see a student make what looks like a careless error and we tell him to check his work assuming he just didn’t think carefully.  Then the student changes the answer because “check your work” is teacher code for “you got the answer wrong.”  Sometimes the error is a case of not being careful.  Sometimes, like here, it’s not.

Instead of assuming it’s a careless mistake, assume that the incorrent answer makes sense to the student and try to figure out why.  Then try to figure out why the correct answer makes sense to you.  The contrast may help uncover the understanding you have that they still need to develop.

Clement, J.  (1982).  Algebra word problem solutions:  Thought processes underlying a common misconception.  Journal for Research in Mathematics Education, 13(1), 16-30.

Clement, J. (2000) Analysis of clinical interviews: Foundations and model viability.  In Lesh, R. and Kelly, A. (Eds.), Handbook of research methodologies for science and mathematics education (pp. 341-385).  Hillsdale, NJ:  Lawrence Erlbaum.

# How to gather the data you really want about student misconceptions

Let’s say you have a binder full of the data I talked about last time.  You know which students got which questions wrong on your latest assessment.

How do you use this information to help those struggling students?

You can’t.  Knowing they answered a question incorrectly is useful, but you need to know why.

That’s where clinical interviews come in.  I like to think of clinical interviews as the diagnostic interview a doctor does when you come into the office.  The doctor doesn’t try to fix what’s ailing you without first making sure he or she has correctly diagnosed the problem.

We need to do the same with students. We need to understand how they are thinking about the math before we can try to help them.

Clinical interviews are used in research to gather data about how a child is thinking.  You give a student a problem and then ask questions as he or she solves it.  The objective is not to teach students but rather to collect information about how they are thinking so that you can later determine the correct remediation.

I know that it’s not realistic to do this with every child in your class.  However, you can learn a lot by trying to incorporate them at times with certain students.

Try this experiment:

Look at your spreadsheet of data.  Take one child who is struggling and try to figure out why.

Give them a task from the assessment (you will need to be careful about how you choose this task) without any answer choices and ask them to think aloud as they work on it.  This may be uncomfortable for them at first since they aren’t used to making their thinking explicit.

You can prompt them to think aloud by asking them:

• Why are you doing _____?
• How do you know ______?
• Tell me more about what you just did.
• What are you thinking about?

As they talk, listen carefully and try to find out what thinking is producing the misconception they are displaying.

A main assumption that I work with when doing these interviews is that children do what makes sense to them even if it seems like nonsense to me.  My job is to figure out what makes sense to them and why.

The more of these you do, the better you will get at seeing things through the eyes of your students. You will start to anticipate the mistakes student will make. You will start to anticipate what might be problematic about certain models you are using. You will begin to discover that the meaning you see in manipulatives or diagrams is not necessarily why they see.

After you have diagnosed what is going on, you can better plan what to do next.  I’ll talk more about ways to analyze the data you gather from these interviews another time.

Want to know more? Check out:

Buschman, L. (2001). Using student interviews to guide classroom instruction. Teaching Children Mathematics, 8 (4), 222-227

Ginsburg, H. P., Jacobs, S. F., & Lopez, L. S. (1998). The teacher’s guide to flexible interviewing in the classroom: Learning what children know about math. Boston, MA: Allyn and Bacon

# Data data everywhere…

During school visits, I rarely make it five minutes without hearing about “the data.”  I am shown big binders containing spreadsheets of data.  I sit in data meetings where teachers furiously highlight lines in spreadsheets and calculate percentages.  Principals tell me how data are being used to evaluate students and teachers.

Some of this is great.  I’m all for assessing children regularly, being clear on what they know and then using the results to decide what to do next.  I would be willing to bet that good teachers have been doing this for years without spreadsheets and binders and meetings.

I also think it’s good to systematize this school-wide and to learn good practices for looking at student work and how to use data to analyze and improve your teaching.  Sadie Estrella is doing some great work on helping schools do this.

However, I think there is a VERY important piece that gets often lost in a rush to analyze data and use results to plan instruction.

Data are only as good as the assessments used to gather them.

That’s one of the most important things I’ve taken away from being researcher.  As researchers, we can spend just as much time designing the tools to gather our data as we do analyzing them.   If the wrong tool is used, it completely changes the conclusions we can make.

A lot of data I see collected in schools are from multiple choice tests that are supposed to reveal who mastered a standard and who didn’t.

There are two potential problems with this.  One is that there is an assumption that choosing the right answer means a student understands the standard.  That may not be true.  Did the student answer the question correctly by guessing?  Did the student answer the question incorrectly because he or she didn’t understand what the question was asking?

The second problem is that knowing how many students in your class answered a question incorrectly is important, but knowing why they didn’t answer it correctly is more important.   It’s impossible to tell from an excel spreadsheet why some students are struggling and others aren’t.

So what tools might help us figure out why students are struggling?

In my line of research, we do a lot of clinical interviews, where we ask students questions as they are solving a problem.  I find data collected that way far more useful in figuring out why students are struggling.   I’ll talk more about clinical interviews and how to use them in your class next time, but if you want to know more now check out the resources below.

It’s important to collect data on our students and analyze what we collect, but we need to spend just as much time thinking carefully about the tools we use and what kind of information we are gathering.

Buschman, L. (2001). Using student interviews to guide classroom instruction. Teaching Children Mathematics, 8 (4), 222-227

Ginsburg, H. P., Jacobs, S. F., & Lopez, L. S. (1998). The teacher’s guide to flexible interviewing in the classroom: Learning what children know about math. Boston, MA: Allyn and Bacon.

# Equality: Part 2

In my last post, I suggested a way to build on what students already know to introduce the equal sign.

Let’s say you followed that plan and your students are now proficient in using symbols to describe a quantity as equal to, greater than or less than another quantity.

What do you do next?

Use numbers. Instead of writing “length of line A=length of line B,” have them measure the quantities and then write number sentences like 8=8 or 7<8.

After enough practice using numbers, give them tasks that invite them to use different operations to create equal quantities.

What might this look like?

Give students Unifix cubes and ask them to complete the following tasks:

• If you have a bar that is 5 units long and you join it to a bar that is 10 units long, is it more than, less than or the same as a bar that is 15 units long?  Show with the cubes how you know.
• Is a 13 unit long bar more than, less than or the same as a 6 unit long bar joined to 7 unit long bar?  How do you know?
• Is a 2 unit long bar joined to a 6 unit long bar, more than, less than or the same as a 3 unit bar joined to a 5 unit bar?  How do you know?

In order to solve the tasks, students can join the bars, determine how large the new bar they created is and compare it to the other bar.   Also include tasks where the quantities aren’t equal.

You can then introduce notation to help students record these relationships in various ways:

• 5 +10=15
• 13=6+7
• 2 +6 =3+5.

Eventually you can move to doing these tasks without the cubes and using larger numbers.

Can you see how this plan could be adapted for other operations?

The key is providing situations that allow students to create different quantities, explore what is the same about them and then record this relationship using mathematical notations.  It builds on what students know and formalizes it using symbols.

I’d love to hear what happens when you try it with students.

This plan was based on the Measure Up research that Barb Dougherty and her colleagues conducted in Hawaii.  It’s a really interesting project and I’ll write more about it some other time, but if you want to know more now, check out the research:

Dougherty, B. (2007). Measure up: A quantitative view of early algebra. Algebra in the early grades, 389-412.

Dougherty, B. J., & Venenciano, L. C. (2007). Measure Up for Understanding: Reflect and Discuss. Teaching Children Mathematics13(9), 452-456.

# Equality: How elementary school teachers can help algebra teachers

I’m certainly not the first person to talk about the importance of the equal sign.  Researchers have been talking about it since the 80’s and discussions about the equal sign have made the rounds on the blogs and twitter (most recently Justin Aion posted about strings of equal signs here)

Often, the concept of equality comes up because high school algebra teachers notice all sorts of wacky things going on when students start working with equations.

Why?

It all comes back to what students understand about the equal sign.  The research on equality shows that understanding that two sides of an equation have the same value and are interchangeable is linked to students’ later abilities to solve equations and reason algebraically.

However, young students are often only introduced to the equal sign as a signal to “do something” or “put the answer here.”  We know that this leads to misconceptions such as answering 12 or 17 to the following problem 8 +4=___+5 or finding the equation 8=8 to be nonsensical.  This causes big problems later on in algebra.

Now, if you’re a high school teacher, check out the comments section of Justin’s post.  In particular, check out what Christopher Danielson  and Sadie Estrella  suggest for how to assess if your students have this misconception and then what you can do to address it.

But I want to talk about what we can do as elementary school teachers that will really help out high school algebra teachers.

Here’s my advice:  Build on what students already know about equal and unequal quantities to introduce the equal sign.

Students have an intuitive notion of more than, less than and the same.  For example, young students know whether their brother is taller than their sister or if two people have the same amount of juice.

Use what already makes sense to students to introduce the equal sign.

I’d suggest starting with quantities first and not introducing numbers at all.  For example:

• Show students two glasses full of juice and ask if glass A has more juice than B, less juice than B or the same amount as B.
• Show them lines of various lengths and ask them if line A is longer than B, shorter than B or the same as B.
• Show them a balance beam and ask them if A weighs more than B, less than B or the same as B.

After a lot of experience examining equal and unequal quantities, formalize the relationships they already know about equal amounts with the equal sign.   Introduce the equal sign as a way to symbolize these situations that involve same amounts.  Then do the same with the inequality signs.

Next, have students write statements such as “length A < length B” or “container B =container A.”  Switch the order they write these statements and ask them to justify whether the statement is still true.

Later, you can do tasks that involve adding or subtracting quantities.  Then you can bring in numbers.   Make sure they understand what the equal sign means first.  Trust me, high school algebra teachers will thank you.

I’ll talk more about tasks that incorporate numbers and operations later this week, but if you have ideas now, please share them below.

If you want to know more, check out the research on equality:

Alibali, M. W., Knuth, E. J., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2007). A            longitudinal examination of middle school students’ understanding of the equal sign and equivalent equations. Mathematical Thinking and learning, 9(3), 221-247.

Dougherty, B. (2007). Measure up: A quantitative view of early algebra. Algebra in the     early grades, 389-412.

Falkner, K.P., Levi, L., & Carpenter, T.P. (1999).  Children’s understanding of equality: A foundation for teaching algebra.  Teaching Children Mathematics, 6, 56-60.

Kieran, C. (1981). Concepts associated with the equality symbol. Educational studies in   mathematics, 12(3), 317-326

# 4 ways to improve classroom conversations

Although I loved the first day of school, September was always a difficult month for me as a teacher.  It’s a month of establishing classroom norms–the rules and expectations of your classroom.   What it means to have a conversation about math in your class may be different than what it means in another class.

As you begin to negotiate the norms in your class regarding classroom conversations, I wanted to share some insight from the research about how to make norms you may already be familiar with even stronger.

Two researchers (Kazemi and Stipek, 2001) observed four classrooms that encouraged classroom conversation about math.  However, the quality of the math talk in the classes differed.  Some classes had norms that encouraged a “high press” for conceptual thinking, while others did not.

Here are four things they observed about how to promote conceptual thinking:

1. Explanations:  While both types of classes asked students to explain their work, high-press classrooms set the norm that explanations needed to be mathematically based.  Repeating a rule or citing a textbook or prior teacher did not count as an explanation.
2. Multiple Strategies: While both types of classes, encouraged students sharing strategies for solving a problem, high-press classrooms focused on what was mathematically similar and different between the strategies.
3. Student Mistakes:  While both types of classes created a culture where it was ok to make mistakes, high-press classroom used mistakes as an opportunity for students to try out other strategies or explore contradictions. The teacher did not validate or invalidate the solutions, but presented both and then asked to students to determine which was correct and why.
4. Collaboration:  While both types of classes encouraged students working together, high-press classes focused on both individual accountability (every student must be able to explain the group’s solutions) and coming to agreement as a group through mathematical argument (as opposed to voting on which they liked best).

As I mentioned previously, change takes time.    It will take a while for students to become comfortable talking about math in these ways.

What do you think?  What norms do you set for classroom conversations?

Read more here:  Kazemi, E., & Stipek, D. (2001). Promoting conceptual thinking in four upper-elementary mathematics classrooms. The Elementary School Journal, 59-80.

# Making New (School) Year’s Resolutions Stick

I love the first day of school.  I love sharp new pencils and blank notebooks.  I love meeting new students that will change your life in ways you can’t even begin to imagine.   Most of all, I love the hope it presents.  It’s a chance to do things differently.

Many of you are excited to try new things this year that you learned about over the summer.  Maybe you attended a workshop.  Maybe you are using a new inquiry-based curriculum.  Maybe you took Jo Boaler’s course on “How to Learn Math.”  Maybe you have been reading the blogs.  Maybe you have been reading the research.

In any case, you are ready to start the school year with a more student-centered approach to math.   Teaching with less of an emphasis on memorizing procedures and more of a focus on developing an understanding of the math concepts.

Here’s the thing.  It’s really hard.  And so the year starts off with the best intentions and then they fall by the wayside come October.

It reminds me of my New Year’s resolution to work out more.  Last January, I took a kickboxing class.   The first thing the instructor asked us to do was ten push-ups.  The problem was I couldn’t do one push up. Not one.  While everyone around me did perfect push-up after perfect push-up, I lied on the mat completely frustrated.

So what did my trainer do?   Did he tell me to give up and go home?  Of course not. He told me to try a push-up on my knees.   That first day, I could barely do three. Then the next week I could do more.   A few weeks later, he had me get off my knees and do a push-up where my legs were far apart.  Each week, I moved my legs closer together until one day I finally did one push-up in perfect form. It took months.

I share this because real change takes time.    There is no quick fix when it comes to teaching and learning math.

Many students aren’t used to a math class that involves problem solving.  They aren’t used to the idea that struggling with tasks can be a good thing.    They aren’t used to looking for multiple solutions.  They aren’t used to justifying their answers.  They aren’t used to having classroom conversations about why a procedure works or whether it works all the time.

If you try to completely change the way students do math all at once and expect change to happen overnight, the kids are going to lie on the mat frustrated.

Just like I needed to build the muscles in my arms and core, your students need to build the muscles in their brains when math class no longer involves the teacher showing them what to do to and helping them as soon as they get stuck.

Building this new culture takes time.  Developing new norms in your classroom will be an on-going process.  Don’t try to do everything at once.  Students may not know how to approach a problem when they aren’t told what to do.  They may not know how to talk about math or what an explanation is or how to justify their answer.

That’s ok.  You will get them there as long if you don’t give up.  Start by adapting problems so they have multiple entry points. Give them smaller tasks at first that will build their stamina.  Then move to more complex problems.  Start with little bits of math talk and have the conversations get a little bit longer each week.  Most importantly, give them time to adjust to a new way to learn and do math.  One day, they will be able to do a push-up in perfect form, but it’s not going to be right away.

I’ll talk more about the specifics of how to begin to do this in another post, but if you want to know now, check out this post and research below.

Carpenter, T. P., & Lehrer, R. (1999). Teaching and learning mathematics with understanding. Mathematics classrooms that promote understanding, 19-32.

Wood, T., Cobb, P., & Yackel, E. (1991). Change in teaching mathematics: A case study. American Educational Research Journal, 28(3), 587-616.

Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for research in mathematics education, 458-477.