Blog #3

Incorporate hands-on activities, manipulatives, and visual aids to help students visualize abstract math concepts.   Walk into most elementary math classrooms, and what do you see?  Containers filled with colorful manipulatives like attribute pieces, coins, clocks, fraction bars, and geometric shapes.  Wonderful tools that help students to make the leap from concrete play and exploration to abstract mathematical concepts. 

Walk into most secondary math classrooms…where are those same wonderful tools to allow students to explore the concrete in order to understand the abstract?  In my experience, there has been a stigma around using manipulatives in the secondary math classroom.  It’s considered too “basic” or too “elementary”.  Why?  Did the math concepts suddenly get less abstract so that students don’t need to “play” with the ideas first?  Do the students already understand everything they need to know and therefore making the leap from concrete to abstract is no longer necessary?  I would answer with a resounding, “NO!!” to both questions.  If anything, concepts become more abstract and students often have more difficulty understanding when we jump straight into the abstract without a little concrete exploration first.  That is why I say, “Phooey!!” to the stigma and find fun, creative ways to use mathematical manipulatives at any and every grade level from K-12 (and beyond).  If you are feeling a little hesitant, what might be holding you back??

• Hard to find for secondary  It is true that if you go to the school supply catalog or website you will have a difficult time finding a pre-packaged set of manipulatives that are helpful at the secondary level.  But don’t let that deter you.  A manipulative does not need to be part of commercially packaged set.  It only needs to be something that students can “get their hands on” to explore different scenarios related to a math concept.  Here are a few simple manipulatives I have used in my classes.

  • • Graphing Calculator  Most high school students have access to a graphing calculator (either their own or a classroom set).  If you use Texas Instruments, they provide a huge library of activities that can be completed using a TI-84 or TI-Nspire.  Most activities are exploratory in nature and include the student activity and teacher notes (as doc or pdf files).  Activities span middle school through college level, so you are bound to find something that will align with what you are teaching.

    • Online Manipulatives  Websites like NCTM’s Illuminations or Mathsbot.com have digital manipulatives that you and your students can use for free.  Desmos.com also has an abundant library of resources that can be used beyond just the graphing calculator application.  Although you can find some activities at teacher.desmos.com, the best way to get wider variety is to open a search engine, type the math concept you are looking for, and add the word “Desmos”.  You will find templates, explorations and reinforcement activities created by Desmos and by teachers just like you.

    • String  String is an amazing tool.  It can be used to do unconventional measuring, to demonstrate conic sections, or to create a human number line.  I definitely recommend keeping a spool of string on hand.  (If you really want to get crazy, get some wax covered string like Wikki Stix or Monkey String!)

    • Handmade Items  Yes, the fancy things you can buy are nice, and will probably last longer, but if the budget doesn’t allow that, you can always make your own.  I have made Algebra Tiles, game boards, question cards, digital Jeopardy game boards, and more.  If you can buy it from a vendor, there are probably directions on the Internet for how to make a version yourself.

    • Physics Teacher  Go make friends with the physics teacher in your school!  They often have wonderful tools that can be used to demonstrate something in your math class.  For example, I would borrow the PASCO projectile launcher from our physics teacher every time I got to vertical displacement with quadratics or projectile motion with parametric equations.  The students loved testing hypotheses to accomplish a goal that I would set, like hitting a target on the floor.

• Not sure how to use them effectively  You do not need to be an expert to figure out how to use manipulatives to help students understand a concept.  If you think something will work, TRY IT!  If you need a little help getting started, check out my TPT Store – Gordon Global Math.  I have an Algebra Tiles product that will walk you through four different activities to help students grasp the rules related to quadratic expressions (Multiplying Binomials, Factoring Quadratic Trinomials a = 1, Factoring Quadratic Trinomials a ≠ 1, and Completing the Square).  This product includes templates for making your own Algebra Tiles, if necessary.  I also have a exploration activity for graphing sine and cosine functions (Spoiler Alert!!!  It uses STRING!)  More activities coming soon.

• Takes too much time  I agree, finding the time and space for students to “play” with manipulatives and explore concepts may be difficult within your schedule.  However, in my experience, I would counter that concern with two observations.  1) The time spent on the exploration usually saved me time later not having to repeat and reteach multiple times.  Students generally understood better and reteaching was less necessary.  2)  Using manipulatives and allowing exploration does not have to be a full-period endeavor.  Sometimes it is, but sometimes it can be as simple as letting students look at a deck of cards when you are talking about probability.  Manipulatives are also not something you will use every lesson, just find a few times when it makes sense and you can make it fit.  The students will benefit even if manipulatives can only be integrated into the lesson now and then.

• What will other teachers say?  Who cares?  If you are helping students better understand math concepts, negative comments from teachers who are jealous or do not understand do not matter.  Besides, I found even the skeptics in my department usually ended up asking if they could borrow some of my “toys”.

• What will students say?  Honestly, I cannot remember a student who reacted badly to using manipulatives.  Even high school students still enjoy “playing” a little, and you are giving them permission to do so as a way to learn.  If you do have a student say something negative, just let them know that this is one tool for helping students learn.  If he/she does not need this tool, that is fine, but ask them to be respectful to those who find it helpful.  (There is a good chance they will join in eventually because it looks fun!)

I hope that this will inspire you to look for ways to bring manipulatives into your teaching now and then, or to increase the frequency that you already use manipulatives.  Not only will it help students learn abstract concepts more concretely, it can also be a lot of fun.  Until next week…email me if you have any questions or suggestions!