Anticipating Questions When Teaching

Do you anticipate questions students might ask as you are preparing to present a lesson?

If not, please do.  It helps students so much!

In thirty-seven years of teaching middle school math, I have probably seen every type of mistake and misunderstanding possible.   

I carefully plan the examples I use when giving notes, trying to include problems in which the most common mistakes students are likely to make are demonstrated.

Think back to your own experiences in math classes.  You will probably remember trying to complete assignments that you thought would be easy, only to find out that the problems at the end didn't look like any of the examples.  

Making connections and applying what you have learned is an important skill.  However, students must have an understanding of how to approach solving a problem in order to apply their knowledge.  We don't want them to become so frustrated that they don't attempt the problems.  This isn't an effective lesson.

Be sure to model your thinking both visually and verbally as you explain lessons.  Students need to see and hear the reasoning and thought processes used when solving problems. 

As you discuss a topic in class, here are some things you can say that may draw students' attention to important details:

• Tell me how we solved the last example.
• How is this problem similar to others we've solved?
• How is this problem different from others we've solved?
• Watch out for this situation ...
• When you see this, be careful to ...
• This is a mistake students often make when solving problems like this ...

Making Connections to Prior Knowledge

Sometimes the little things you say or do as a teacher that can be most helpful to students.  Make connections by linking what students already know to new information.

Activating prior knowledge sounds like a great idea in theory, but do you actually do it?  In your college methods course, it was sure to have been emphasized as a strategy we should all be using.  

I'm not one to create an elaborate introductory lesson leading into a new topic.  You could do that, but how you introduce a topic may be as simple as showing students one example of a problem they may have been expected to solve a year or two earlier, then following up with an example of what the expectation for the current year looks like.

Example #1

If I'm teaching algebraic proportions to my algebra class, I may show them a typical proportion similar to the one below.  Then I would go on to say that the second example is also a proportion.  Therefore, you would solve it the same way, by finding the cross products because cross products in any proportion are always equal.

It can be that simple.

It is tempting to jump right into the new lesson, but just one quick example can turn the light of understanding on for students.  

Example #2

Suppose you are introducing the topic of writing consecutive integer equations.  Rather than immediately telling students how to represent each number, begin with the basics.

Ask, "What are consecutive integers?"

If there is no response, follow up with, "What does it mean to be at school on consecutive days?"

Someone should be able to answer that question and you can use that response to lead into the idea of consecutive integers.

Also, you might ask what is meant by consecutive evens and consecutive odds.

Below, see a sample script of follow up questions to guide students to a better understanding of the problem set up.

You've activated prior knowledge through this simple discussion.

Example #3

When teaching students to solve inequalities, have them solve an equation such as the one shown below.

Next, change the equal sign to an inequality symbol.

Tell students to pretend the greater than or equal to symbol is an equal sign.  As long as they don't flip the inequality over as shown below,

the steps involved in solving the inequality will be just like solving an equation until the last step.

Stress that the only new rule they are learning is to reverse the inequality symbol in the last step if they multiply or divide both sides by a negative.

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Students worry about the multiple steps often involved in math problems.  This can cause some of them to shut down right away when beginning a lesson.  Building self-confidence in mathematics is so important!

Linking the topic to a skill or concept that students already understand can build confidence and more readily translate to success.

By clearly establishing at the outset of a lesson, that most of the material is review and only a small part of it is new, you can motivate them to keep working and learning.

Creating a positive atmosphere in your classroom in which students feel encouraged and are developing self-confidence can make a huge difference in their attitudes about the subject in general.  It starts with making connections to prior knowledge.

Do I really need to teach that? It’s common sense!

What you think is common sense may not be for your students,.  Remember, you are an adult!  While you may think they should just know certain things and that students will make connections using number sense, it is doubtful that they all will.  Teach even what you believe should be common sense.

Wouldn't you be doing your students a disservice by not drawing their attention to those simple connections that could really help them?

Here are three examples based on my experience in the classroom:

Example #1

When graphing linear equations, students don't typically look at equations such as x = 3, y = 8, and 3x + y = 6 and think about what type of line each will form.

So, they just graph a line somewhere on the grid without considering whether the line should be diagonal, horizontal, or vertical.

Fixing this issue can be as simple as having students predict how a line will look before putting pencil to paper.

• Have them write "horizontal", "vertical" or "diagonal" before graphing.
• Give a quick check for understanding focused on only this topic.
• As you work through examples in class, ask students to predict what type of line to expect before graphing.

I think of this as having an understanding of the "big picture".  Others describe it as common sense.

 

Example #2

Another topic I've noticed that students struggle with is labeling perimeter, area, and volume.  Many of them just don't make the "big picture" connection.  They perform all sorts of contortions to decide how to label an answer.  Some seem to think that the shape determines the labeling.

All they need to know is this:

• All areas are square units (area, surface area, lateral area, base area).
• All volumes are cubic units.
• Anything else is single units (length, base, width, height, perimeter, circumference).

It is not common sense ... teach it!  Every student will not just figure it out.  If so, I would need to be writing this post.  

Example #3

The same goes for positive vs. negative slope.  A positive slope is a line that slants upward from left to right showing an increase.  A negative slope is a line that slants downward from left to right showing a decrease.  Train your students to think about whether the slope will be positive or negative before counting it or graphing a linear equation.

Verbalize the key "big picture" takeaways.

Remember, common sense is not common!

How to Introduce Negative Exponents

Teaching lessons where patterns are involved is fun!

When I began having students make charts as they were learning about negative integer exponents, it made it so much easier for them to understand rules and concepts.

Let’s face it - to students, it's not logical that negative exponents result in fractions.  Looking at a chart and examining patterns changes promotes understanding.

I begin by having students complete a chart like the one below.

Next, I have them extend the chart as shown.  Then, ask students to study the chart and explain what is happening as they examine the chart which goes from 625 to 125, 125 to 25, and 25 to 5.  Hopefully, they'll recognize that each number is divided by 5 as they work their way down the chart.

I always tell my students that math is all about patterns. 

They love to look for patterns!

We complete the rest of the chart together by continuing with the pattern of dividing by five.  This is a perfect way to review the exponent of zero rule and introduce negative exponents at the same time.

After completing this chart with students, I have them complete a similar one on their own.  Then, I follow that up with a brief class discussion to reinforce the observation that negative exponents result in fractional answers.

Parentheses when Substituting

Do you sometimes think, "I wish my students could just remember this one thing?"

I'd like mine to remember to use parentheses when substituting into an algebraic expression.  I stress outlining the problem before replacing the variables with numbers.  However, I can't force students to do it.  It can be frustrating because if they do outline the problem first, as shown below, they generally solve it correctly.

One little trick I use when working examples for the class is to highlight each space where a number will be substituted.  Little things like this can help put the focus where it belongs, and is especially helpful to students who are visual learners.

Big Picture Equation Solving

 Do you have students who get lost while solving a multi-step equations?

Does it appear that they are randomly writing work without understanding what to do when or why?

A different perspective is needed when this happens.  And ... it can be simple!

In the midst of solving a multi-step equation, students can lose track of the goal - to solve for the variable.

What I call "outlining the equation" is one technique I use in this situation.  It doesn't help all kids, but it does help a number of them.  It is a visual that keeps the end goal in sight.  Students begin by making an outline as shown, using DCV.  

Distribute - Combine Like Terms - Get all Variable Terms on One Side

After completing DCV, it looks like this:  

Success!

Algebra: Teach the Instructions

Have you noticed this when teaching Algebra?

I’ve found that when my algebra students are struggling, the cause is often not the math, but the instructions and vocabulary.

For example, if the instructions say “factor the polynomial completely”, students often write solutions as if solving an equation.  As shown below, the student knows how to factor, but goes beyond that due to misunderstanding how an expression and equation are different. 

Often it seems as though students are understanding the material, yet they don’t perform well on assessments.  The example above shows one reason why this may happen.

Here is another example: “Find the zeros to the quadratic function.” or “Find the roots to the quadratic function.”  (Hands go up … “What are zeros?”  or What are roots?”)

The issue is the vocabulary term.  My students may have mastered using the quadratic formula, yet not have a clue how to begin to answer this question.

How can we fix this you ask? 

When you begin the quadratics unit, teach the vocabulary including terms students may see in instructions.  Stress that “answers, solutions, roots, zeros, and x-intercepts” are all synonyms.  Because you know what?  It is confusing!  Although you may not typically consider these to be vocabulary terms for the unit, being familiar with these synonyms is critical to success.

My advice is to make a conscious effort to teach students about not just the algebra, but about the most common errors they are likely to make and how to understand instructions.  You may think it is common sense, but kids don’t have the experience teachers do.  After completing high school and college, we’ve solved so many problems similar to the one above that we naturally do what we should because we have learned.  Students are still learning!