10 Things I Wish I Had Known About Teaching Algebra

Summary for Educators

By Jill Newton | Mathematics Teacher: Learning and Teaching PK–12 | September 2026 | Vol. 119, No. 9, pp. 742–747

Teaching Algebra well requires far more than showing students the correct steps for manipulating symbols. It means helping young people understand why algebra works, how ideas connect, how representations communicate meaning, and how to persist when mathematics becomes unfamiliar.

In “10 Things I Wish I Had Known About Teaching Algebra,” mathematics educator Jill Newton reflects on lessons that emerge from experience teaching and studying algebra. Her larger message is especially important for secondary teachers: knowing algebra and knowing how to teach algebra are two different forms of expertise.

THE BIG IDEA

Algebra can easily become a course about procedures: distribute, combine like terms, isolate the variable, factor, substitute, and simplify. Students may learn these routines well enough to complete familiar exercises yet still have difficulty explaining what an equation means, recognizing the same relationship in a graph or table, or deciding what mathematics to use in an unfamiliar problem.

Effective Algebra instruction therefore needs to go beyond teaching students what to do next.

Teachers must continually help students make connections between arithmetic and algebra, symbols and meaning, equations and graphs, procedures and concepts, and individual problems and larger mathematical ideas.

Newton’s work on preparing Algebra teachers reinforces this distinction. Research involving prospective mathematics teachers has found that teachers can successfully perform common algebraic procedures while having much more difficulty explaining concepts through multiple representations or anticipating how students might understand them.

KEY TAKEAWAYS

1. Algebra begins long before students enter Algebra I.

Students' understanding of fractions, proportional reasoning, operations, equality, patterns, and number relationships becomes the foundation upon which formal algebra is constructed. Weaknesses in those foundations do not simply disappear when letters replace numbers.

2. The equal sign deserves serious attention.

Many students enter Algebra thinking “=” means the answer comes next. Algebra requires them to understand equality as a relationship between two equivalent quantities. That conceptual shift affects equations, functions, transformations, and virtually everything that follows.

3. Variables are more complicated than they appear.

A letter can represent an unknown number, a changing quantity, a generalized number, or a parameter. Students need opportunities to encounter these different meanings rather than simply being told that x is the number you are trying to find.

Research-based Algebra teacher-development materials similarly identify variables and relationships as major conceptual hurdles for students.

4. Multiple representations matter.

Students should learn to move among equations, graphs, tables, diagrams, verbal descriptions, and real situations. Each representation exposes something different about a mathematical relationship.

The goal is not merely to have students produce several representations. It is to ask:

“What can you see here that you couldn't see as easily in the other representation?”

5. Procedures need meaning.

Procedural fluency matters. Students eventually need efficient ways to solve equations, manipulate expressions, and work with functions.

But shortcuts taught without understanding can become fragile rules that students misapply.

Instead of only asking, “What step comes next?”, teachers should routinely ask:

“Why is that step mathematically legal?”

6. Student errors are valuable information.

Wrong answers provide a window into student thinking.

A student who writes
3(x + 4) = 3x + 4

has revealed something important about his or her understanding of the distributive property.

The instructional opportunity is not simply to mark the answer wrong but to uncover the reasoning that produced it.

7. Connections are the heart of Algebra.

Strong Algebra students increasingly see mathematics as an interconnected system rather than a collection of isolated chapters.

Newton's research has specifically examined the importance of helping future teachers recognize and develop algebraic connections across concepts and representations.

8. Technology should reveal mathematics—not hide it.

Graphing tools, dynamic mathematics environments, spreadsheets, and computer algebra systems can help students explore relationships that would otherwise be difficult to see.

But technology alone does not improve mathematical understanding. Its value depends on the questions teachers ask and the thinking students do. Research-based Algebra materials from NC State similarly emphasize using technology alongside careful planning, assessment, and analysis of student reasoning.

9. Students need opportunities to explain.

An Algebra classroom should contain mathematical conversation.

Students should regularly explain:

• why an answer makes sense,

• why two expressions are equivalent,

• what a graph tells them,

• why a method works,

• how two solutions differ.

Explaining mathematics often reveals understanding—or misunderstanding—that a correct numerical answer conceals.

10. The teacher never finishes learning Algebra.

One of the most powerful lessons for mathematics educators is that teaching a subject exposes dimensions of that subject that solving problems alone does not.

Teachers continually encounter new student strategies, misconceptions, representations, questions, and connections.

Teaching Algebra well is therefore not mastery followed by repetition. It is a continuing process of studying mathematics through the eyes of learners.

WHY IT MATTERS

Algebra remains one of the most consequential transitions in a student's mathematics education.

It is where mathematics moves increasingly from computation toward abstraction, generalization, relationships, functions, and mathematical modeling. Success in Algebra strongly influences students' access to later mathematics and STEM coursework.

That means schools should resist reducing Algebra improvement to pacing guides, test preparation, or simply giving students more practice problems.

The more important question is:

What kind of mathematical thinking are students developing while they learn Algebra?

LEADERSHIP ACTIONS

School leaders and mathematics department chairs can use Newton's reflections as a useful lens for examining instruction.

During classroom visits, look beyond whether students are quiet, engaged, and getting correct answers.

Ask instead:

• Are students explaining their mathematical reasoning?

• Are teachers examining misconceptions rather than merely correcting answers?

• Are students moving among graphs, equations, tables, and words?

• Are teachers connecting new Algebra concepts to previously learned mathematics?

• Are students asked why procedures work?

• Is technology increasing mathematical thinking—or replacing it?

• Do collaborative planning conversations focus on how students think as well as what teachers will teach?

Professional learning can also move beyond simply sharing activities. Teachers benefit from collaboratively examining actual student work and discussing what that work reveals about students' mathematical understanding.

LEADER REFLECTION

Walk into an Algebra classroom tomorrow and resist asking:

“Are students getting the right answers?”

Instead ask:

“What evidence do I see that students understand what the mathematics means?”

That small change in question may completely change what we notice about Algebra instruction.

Original Article

“10 Things I Wish I Had Known About Teaching Algebra” by Jill Newton in Mathematics Teacher, September 2026 (Vol. 119, #9, pp. 742-747)

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Prepared with the assistance of AI software OpenAI. (2026). ChatGPT (5.2) [Large language model]. https://chat.openai.com