Category Archives: Algebra

Using diagrams to ease the transition to algebra

People sometimes feel compelled to tell me how much they hate math. When I ask when this started, many point to algebra (the rest say fractions, but we can talk about that another day).

We know that many students struggle when they get to algebra. Researchers have been studying how to ease this transition through “early algebra.” They found that simply introducing traditional algebra concepts at an early age isn’t the answer.

Instead, Blanton & Kaput (2005) talk about “algebrafying” the elementary school curriculum, in which elementary school concepts are developed in a way that allows generalizations about properties and relationships to become more explicit.  For example, students are asked to identify and generalize about patterns, relationships and structure in mathematics. They are given tasks that require them to reason about unknown quantities. At early ages, they develop the ability to identify, describe and analyze how quantities vary in relation to each other.

As you may have noticed, I’ve been obsessed with tape diagrams recently.  In fact, I’m doing a Global Math Department session about them tomorrow night if you want to hear more. One of the benefits I haven’t talked about yet is their potential for developing algebraic thinking.

Take a look at this problem.

Nicora wants to buy herself a new bicycle that costs $240.  She has already saved $32, but needs to make a plan so she can save the rest of the money she needs.  She decides to save the same amount of money each month for the next four months.    How much money must she save each month to meet her monthly goal of buying a bicycle?

Many of you probably solved it by setting up the equation $240= $32+ 4x and solving for x.

However, younger students could also solve it by setting up a tape diagram like the one below.


After creating the diagram and using it to solve the problem, students can talk about writing a number sentence to represent the situation. This allows the to give meaning to each part of the equation. Students can discuss what the $240 represents, what $32 represents and what the missing number, x, represents. They can also discuss how they solved for the missing number. In this way, the tape diagram can be used to build developing algebraic thinking.

Want to know more?

Blanton, M., & Kaput, J. (2005). Characterizing a classroom practice that promotes         algebraic reasoning. Journal for Research in Mathematics Education, 412-446.

Asking questions that help students think algebraically

Over the past two weeks, I’ve been working with both students and teachers on algebra tasks. One of the things we have been doing is writing rules for different geometric patterns.

For me, identifying key questions that will guide students towards meeting the learning target for the day is one of the most important things I do when planning a lesson.

I find that it can be easy to do really cool activities that get students engaged and have great potential but if the questions we ask don’t pull out the mathematics, kids get excited about math class but don’t learn much. The questions asked as the students work need to guide them to see the connections and pull the mathematics out of the situation (which may be obvious to us but not to them).

I think sometimes the resistance to inquiry based approaches is because if students aren’t carefully guided to see the mathematical connections, it’s no better (and maybe even worse) than simply telling them a procedure and having them practice a bunch of problems.

When working on my algebra lesson, I went to researcher Mark Driscoll’s book, Fostering Algebraic Thinking: A guide for teachers, Grades 6-10. In it, there are various questions that can be used to foster different types of algebraic reasoning. Below are some of the questions he suggests using when students are engaged in writing rules to represent functions:

Questions for building rules to represent functions

  • How are things changing?
  • Is there information here that lets me predict what’s going to happen?
  • What steps am I doing over and over?
  • How can I describe the steps without using specific inputs?
  • What if I do the same thing with different numbers? What still holds true?  What changes?
  • Can I write down a rule that will do this job once and for all?
  • Why does the rule work the way it does?
  • Does my rule work for all cases?
  • Now that I have an equation, how do the numbers in the equation relate to the problem context?

I used these questions to help assist small groups when they were stuck as well as to guide the whole class discussion. They helped students see the connection between the pattern they were drawing and the algebraic rule that could be written to describe it.

Want to know more?  Check out: Fostering Algebraic Thinking: A guide for teachers, Grades 6-10 by Mark Driscoll

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

Simplifying expressions isn’t so simple

“…some seemingly simple ideas are not always as simple for students as they seem to adults.” -Lesley Booth

How often have you seen students make the following type of mistake?

2a +5b=7ab

My guess is that if you teach algebra, you’ve seen this mistake more than once.

If we believe that students do things that make sense to them, we need to ask:

Why does 2a + 5b=7ab make sense to students?

Here’s what the research has to say:

  1. The nature of “answers” in algebra is different. From their prior experience in arithmetic, they have developed certain expectations of what answers look like. They assume that a “single term” answer is what is needed. There is also evidence that points to the idea that students give one term answers because they have difficulty accepting a “lack of closure.”
  2. Algebraic notation is confusing to students. They don’t necessarily see the invisible operation between the variable and the number as multiplication. If anything, they may assume the hidden operation is addition given their experience so far in math. For example, when we write , it means 4 + 1/2 and when we write 43, it means 4 tens + 3 ones. It makes sense to them that 2+a+5+b would be equal to 7+a+b.
  3. Students sometimes view the variables as concrete objects. For example, they see a as apples instead of “the number of apples” and b as bananas instead of the “number of bananas.” Therefore, they justify 2a+5b=7ab by saying 2 apples plus 5 bananas is equal to 7 apples-and-bananas.

Here’s the more difficult question: After reading why this mistake makes sense to students, what might we do differently in our teaching?

I’ll talk about what the research suggests next time, but if you want to know now, check out the work below.

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

Tirosh, D., Even, R., & Robinson, N. (1998). Simplifying algebraic expressions: Teacher awareness and teaching approaches. Educational Studies in Mathematics, 35(1), 51-64.

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?

Let’s start with addition.

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.


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