Blog #13

Many students struggle when you try to explain to them how the domain of the composition  is not always just the domain of the first function .  By showing this series of function machines, they can visualize why we have to pay attention to the range of the  function and make sure it is going to work with the domain of the  function before we can state the domain for the entire composition.  I tell them that the connection between the  machine and the  machine is enclosed, so the only way they can control what goes into the  machine is to control what goes into the  machine in the first place.  This gives students a concrete place to begin understanding why and how we confirm what the domain of the entire composition should be.

Visual aids can also be applets found online.  For example, using this Geogebra applet, you can visually help student make the connection between the sine values they have identified on the unit circle, and the sine curve that they graph on a coordinate plane.  This type of applet shows students that math concepts can be represented in various ways (numerical, graphical, algebraic) and that knowledge about one representation can help them understand another representation more easily.  This could also be accomplished with a short video.  This YouTube video from TED-Ed is a nice, quick way to show students the concept of exponential growth.  There are so many resources available at your fingertips that it should not be difficult to find images, applets or videos that give students the concrete foundation they need to build on.

One last possibility for visual aids is to use manipulatives.  Things like dice or playing cards to understand probability, or Algebra Tiles to “play” with linear and quadratic expressions.  These tools make the learning fun, tactile, and accessible, especially for students who need a little extra help.  (For more ideas related to manipulatives, please check out blog #3)

Break Down Problems

We need to teach students how to decode, decipher and deconstruct math problems.  This is NOT a skill that we can just assume that students possess.  We as math teachers need to talk students through the process of working through complex problems. 

Mathematical language is one area where students can struggle.  Obviously, if you have second language learners in your classroom, this is imperative.  But even with native English speakers, mathematical language can get tricky.  We need to include lessons on how to “translate” math-ese into language students understand.  When a problem asks you to “find the rate” in Algebra 2, you are likely dealing with an exponential function.  But, when the problem says “find the rate” in calculus, that is likely referring to derivatives.  Students need help interpreting what information the problem provide, and what it is asking students to do.

Smaller, more manageable steps can be a helpful skill for students to learn.  Many students, when faced when a complex, multi-step problems, get lost in the weeds and want to give up.  As we help students decode what the problem is about, we can also help them determine what math skills may be involved.  In the Algebra 2 rate problem mentioned above, once we determine we are working with an exponential function, there are certain characteristics we should identify.  Does the problem seem to be talking about growth or decay?  That will let me know whether my answer for the rate should be >1 or <1.  What amount am I starting with and how much do I have after a certain amount of time?  That will help me set up an equation.  If the problem is actually asking me which population is growing faster, then that tips me off that I will need two equations, and once I find each rate, I need to compare them for a final answer.  This process, which may feel like second nature to us as teacher who have been doing these problems for years, is daunting to a struggling math student.  Asking students strategic questions can get them to talk through the steps of the process.  When students then arrive at a reasonable solution, they will feel more confident to tackle similar problems on their own.

Determining the best approach to a problem is another way we can scaffold learning for students.  Too often, it seems like we teach students one way to approach a problem because of what lesson we are learning in the textbook, or because in algebra class we focus on algebra.  But again, students need to realize there are usually multiple representations for math problems.  For some students, solving an algebraic equation may be too overwhelming, but using a graph to find a solution may be more reasonable.  Or maybe making a table of values is more accessible than writing an equation to solve or graph.  We need to give students permission to use alternative methods to set up their problems.  We also need to encourage them to look for those other approaches that may make solving the problem easier.  If they can build confidence using one approach, we can make connections to other approaches and can help move students closer to the learning targets we want them to meet.

Provide Guided Practice

Students need guided practice.  If we liken learning math to learning to ride a bike, children usually need to start riding a bike with training wheels.  Then they need to take a few tries with an adult running alongside them before they are off and riding on their own.  Similarly, students often need the mathematical “training wheels” of working through a few problems as you demonstrate how to do it.  Then they need to take a couple tries with you or a peer right beside them, offering encouragement and feedback to help them improve.  Only then are students ready to go out on their own and do the practice/homework for the lesson.  Too often, because of the time constraints we must manage, we spend the whole lesson showing students how to “ride the bike”, then give them a bike at the end of the lesson and expect them to take off down the street.  There are a few students who can do that, and I am not suggesting that those students be required to use training wheels first.  If they can ride, let them ride!!  But for most students, we must be intentional about providing practice with someone “running alongside them” before we let them go on their own.  This will help students build up their confidence that they are on the right track and are ready to try out their newly learned skills.

Providing scaffolded activities and opportunities when students demonstrate the need for them will help your students assemble the building blocks to be successful math learners in your class and beyond.  You will be creating a learning environment that fosters student growth and confidence in math.  Students will know that they are valued in your class because you are taking the time and making the effort to help them succeed.  What could be better?!  Hope you all have a great week!!