# 5 ways to make simplifying expressions make sense

Student mistakes are important.  It’s important to know what mistakes students make and why they make those mistakes.

For example, last time, we looked at  why simplifying expressions isn’t so simple for students and why they tend to make mistakes like 2a +5b=7ab.

When we can anticipate what mistakes students commonly make and why they make them, we can began the hard work of figuring out how to develop understanding so these mistakes don’t make sense to students.  We can start to answer questions like the one below.

How do we make simplifying expressions make sense to students?

Here are 5 ideas from the research:

1. Write out the hidden multiplication in algebraic expressions.  We know that students often mistake the missing operation in 2a as addition.  In order to avoid this, write out the expression in expanded form at first (a x 2 or 2 x a).  Give students lots of experience simplifying expressions with this expanded notation at first and then gradually move away from it.
2. Develop meaning for the equal symbol.  I’ve talked about how important this is before and suggested some ways to do this starting in early grades.
3. Don’t use the fruit analogy.  It isn’t helpful.   Two apples plus five bananas equals seven apples-and-bananas makes perfect sense to students.  Plus it creates confusion about a variable represents.
4. Avoid using the same letter for the variable as the first initial of what it might represent.  Although it appears makes sense to us to use a to represent the number of apples, students see it as apples and they start to believe that the variable is a label for the object.   Once students have a clear understanding of what a variable is, you can use the same letter, but try to avoid it at first.
5. Give students experience with substituting values for variables before and after simplifying expressions and encourage them to find counterexamples.  Through continued experience, they will strengthen their understanding of why certain terms can be simplified and others can’t.

I’d love to hear what other suggestions you might have.

Want to know more?  Read the articles below.

Booth, L. R. (1988). Children’s difficulties in beginning algebra. The ideas of algebra, K-12, 20-32.

Carpenter, T. P., & Levi, L. (2000). Developing conceptions of algebraic reasoning in the primary grades. Madison, WI: National Center for improving student learning and achievement in mathematics and science.

MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation. Educational Studies in Mathematics, 33, 1-19

# 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.

# 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