Monday, December 26, 2016

Manipulatives in Secondary Math

As a high school math teacher, in the classroom, I made very little use of concrete manipulatives such as cube links, square tiles, or the ever-dreaded Algebra Tiles.

I say ever-dreaded because while Algebra Tiles have long been touted as an amazing resource, it's never been obvious to me how to use them. 

If you haven't seen or used them, they are a collection of small squares that represent units, large squares that represent x^2, and rectangles (with a length that matches the large square, and a width that matches the small square) that represent "x." Students can use various combinations of the shapes to model equations and algebraic thinking, leading up to more formal mathematical processes.

Other than their most basic uses, they confuse me. I don't think like that (visually), and I certainly wasn't taught like that. 

I did well in math through high school because I like rules. I could memorize and apply the rules for solving a linear equation, or factoring a quadratic equation, or completing the square. Though I may not have understood the math at the time (I was just following rules, remember), with lots of practice I was eventually able to see why the rules worked, and could apply that line of thinking to solving even more complex problems. 

So I didn't make good use of manipulatives as a teacher partially because I never really learned how, either through experiencing it as a student, or by experimenting with them as an adult. 

I'm changing my mind...

But what I'm learning this year, is that not only are these concrete manipulatives a good option for differentiating our math instruction, they are a NECESSARY instructional tool. 

For our students presenting with learning disabilities, there are a number of reasons why using manipulatives regularly in the classroom is beneficial. Among others, these include:
  • For students demonstrating slower processing speeds, manipulatives force the pace of learning to better match that of the student
  • They provide a means for students with working memory needs to better keep track of what they are doing, by displaying the process on the table in front of them. 
  • They allow students to make use of perceptual reasoning skills to accommodate for needs in mathematical computation.

However, for ALL our students, manipulatives provide a depth of learning beyond what I was exposed to as a student. I was never taught how to think of algebraic processes outside of just rules for making numbers appear and disappear. I wonder how much more quickly I would have seen patterns and made connections if I could have visualized what the equations represented? 

There is a stigma associated with using manipulatives in high school - that they are only for the kids that "can't do math." But what if their use isn't seen as a "crutch," but strictly as another way of thinking/seeing the math (which is exactly what they are!). Students who feel they don't need manipulatives (like I once was) could actually be encouraged to think mathematically in a new way.

This is something I need to start doing more of.

This isn't an easy transition for me - building manipulatives into my arsenal of teaching tools is going to take a lot of learning (and playing?) before I'll be comfortable with them. But for now, I can at least envision what this might look like. When I go back to the classroom, I'll be aiming to:

  • Physically move the manipulatives INTO my classroom (out of storage) and have them in an easily-accessible spot for everyone to get to, not out of sight in an office or tucked away in a classroom closet.
  • Incorporate manipulatives purposefully into lessons - carefully choose which manipulative the students will be using and know why I'm choosing to use it. What process does it demonstrate? In what way will it help my students think/reason?
  • Make manipulatives integral to the lesson itself, not just have it as an add-on to what we're learning. 
  • Challenge the students to whom math comes easily to use the manipulatives, and get them thinking outside of the memorization box. I hope this might also reduce the stigma of using manipulatives.

But I still wonder...

  • How well do digital manipulatives benefit students (if at all)? Are apps worth investigating?
  • What resources are available for getting good, challenging-yet-accessible activities with manipulatives at the secondary school level?
  • How do students learn which manipulative to use when presented with a selection (or when they can choose what to use on standardized tests)?
  • How can I best incorporate manipulatives into flipped lessons?

I'd love to hear of good resources already in use out there for activities and materials that actively engage high school students in learning algebraic processes, and try my hand at some of them!


  1. Digital manipulatives: in my experience, they don't generate the same level of student interaction as physical manipulatives. On the other hand, they're easier to manage!

    Resources: check out my Web site, where I have a ton of materials on manipulatives for secondary school. Maybe start here:

    Good luck! And feel free to get in touch if you have questions. I have decades of experience with this.

    -- Henri

    1. Hi Henri - thank you so much for sharing your resources! I'm looking forward to exploring them more :)

  2. Hi Heather,
    After a long time out I am back in the classroom trying many of the things you are talking about here. Here are a couple of things I've been trying so far...
    1) In the spirit of transforming learning spaces, I now have a series of long tables down the centre of the classroom with various manipulatives on them. Students sit (when they are not up and moving) in 4-desk pods on either side of these learning tools. Management strategy: Lids off means these tools are fair game for what we are learning today. Lids on, means leave them alone for now.
    2) I use virtual manipulatives as a companion learning tool for the real ones, not necessarily a replacement. Example: When investigating linear growing patterns using link cubes, square tiles, etc. (student choice), I can share/project models using 'real versions' by holding them up or placing under the document camera. I can also use a virtual version, such as the CLIPS square tile tool. An advantage of the latter is I can efficiently copy/extend patterns and change the colours as we discuss the gradual release of colour coding of the variable and fixed parts of a growing pattern. I also encourage students to use the virtual tile option on their BYODs as one of their manipulative choices.
    I have a long way to go in figuring out all the best strategies, but this is where I am at right now...