Suggestions For Improving Algebraic Knowledge

 Most of the time, we teach in ways that are not suited to help students improve their algebraic knowledge.  The current system is digitally based but it still has the examples set up so you click the screen and each step is shown in the proper order.  The students work their problems and are expected to learn but this method is not necessarily the best way to do it.

In 2015, The U.S. Department of Education made several suggestions designed to help students improve their algebraic knowledge.  At least one of these is contrary to what I learned when I was in my teachers education program but it makes a lot of sense.

1.  Start with a problem that has already been solved and shows all the steps.  You want students to analyze the problem, the process used to solve it and let them make connections among all the strategies and reasons they've learned.  You need to make sure the problem is one that focuses on the lesson's instructional goal.  You should include problems that show common errors so students become proficient at recognizing them.  Be sure to use a variety of methods such as small groups, whole groups, pairs, etc to discuss this.  This example could be pulled from the book, student work, or could be made by the teacher.

It is important to help students learn to analyze problems by having them describe the steps used to solve the problem, asking questions such as "Could it be solved in fewer steps?" or "Are there other ways to solve this problem?" or "Will these strategies work for other problems?  If so which ones?".  Furthermore, it is important to use more than one example and the examples should have different levels of complexity so they can see they cover the same concept.  By using several problems, students are able to see the steps are the same for all problems.

When having students analyze problems that were done incorrectly, let them verbalize why the error lead to a correct answer.  Another way is to have both the correct and incorrect problem next to each other so students can compare and contrast the steps to find the incorrect step.  It is important to ask probing questions such as "What advice would you give the student to help them understand why they did it incorrectly?"

2.  Encourage students to use algebraic representations by using language that promotes said mathematical structure. Teach students to use a type of self reflection questioning as they solve problems to help them see structure,  and help them see that the different types of algebraic representation can help them see different types of information.

Structure refers to the type and number of variables, operations, equality or inequality signs, and relationships among all of these.  It is important to use precise mathematical language so students develop the vocabulary and the connections between words and structure.  Furthermore, it is important to take time to teach students some questions they can use every time they solve a problem so they recognize structure.  The questions might be something like "What can I say about the form of the expression" or "What has happened in similar problems before".

It is important to teach students to compare and contrast different forms to what they focus on.  For instance the "Slope intercept form" makes it easy to graph starting with the y-intercept while the "Point slope form" begins at a specific point on the line.  Both forms have their uses depending on what you need to do and what you have to work with.

Furthermore, it is necessary to teach students to represent problems, especially word problems, using different methods.  Sometimes it helps to teach them to translate a word problem into a specific visual representation so they see what is going on.

3.  Take time to suggest students look for alternative ways to solve problems.  To do this, the teacher needs to help students learn to generate a list of possible ways to solve problems, look at the pros and cons for each method, and explain their reasoning for choosing a particular method.

One way to accomplish this is to show students different ways to solve the same problem including the standard algorithm that is usually taught.  By showing different ways of solving the same problem, students have the opportunity to see which ones might be more effective than others.  It is also important to let students come up with strategies on their own to try.  Furthermore, students should see how the same strategy can be applied to different problems so they see a connection.

Do not use all the alternate strategies at once or you might overwhelm the students. Introduce one or two at a time so they can practice them.  Teaching students a variety of strategies can help them become better at approaching new types of problems they have not seen before because they have tools to work with.

It is important to question students throughout the whole process to help them develop their self reflection and ability to try new problems.  Let me know what you think, I'd love to hear.  Have a great day.