Three Conceptually-Based Maths Activities That Don’t Require Hands-On Equipment

Educators often assume conceptually-based mathematics activities for high school students always require lots of hands-on equipment. It's a misconception, by the way, which I'll explain later.

In an ideal world, high school maths students would explore hands-on-materials, within a well-structured setting, in a highly-engaged, inquiry-based, self-directed manner.?

However, last I checked, the world is not ideal, and MOST maths teachers resist activities based on hands-on-materials, for three reasons:

• Hands-on activities in many high school classes are challenging to manage.
• Students manipulating hands-on materials, in many cases, does not correlate to the students being mathematically switched on. To quote the late Grant Wiggins, 'Just because it’s hands-on, doesn’t mean it’s minds-on’ (Experiential Learning ).
• Hands-on activities often require more time than traditional teacher-directed activities.

To draw on personal experience, I’ve run several potentially conceptual-rich lessons that fell very short of their potential. Some students failed to engage, and most of those who did engage failed to make the intended mathematical connections. And the students were mostly on-side.?

Let’s be clear, I’m not against students using hands-on equipment. I’m merely stating that for many high school maths classes, using hands-on equipment in a time-efficient way that engages students and has them making the intended mathematical connections - is unrealistic.?

Misconception: Conceptually-based activities require hands-on equipment

Have you ever wondered what it is about a (well-structured) hands-on activity that makes it conceptual in nature???

Ten seconds to ponder … dum-dee-dum …?

OK, times up!?

My take on why hands-on activities are conceptual is this: There is no procedure to follow!

If an activity has no procedure to follow, it forces students to explore the underlying concept. And the beauty of activities that 'force' students to explore the underlying concept is this: if students do not understand the concept they CANNOT produce correct answers.?

And this is EXACTLY what we want - a direct correlation between correct answers and understanding the related concept.

Compare this with the traditional approach of teaching procedures and having students replicate them - many students attain correct answers without understanding what they are doing. They don’t understand the underlying concept.

(Note here, that we are not advocating the teacher explain - in words - the concept or the required understanding. Some explaining is necessary as a summary AFTER students have grappled with the concept themselves. Even then, it is better that students explain their understanding before the teacher summarises it!)

Create activities that have no procedure to follow

Therefore, if we want students to understand their maths work - i.e. understand the concepts that underpin the procedures they are working with - then we need to give them conceptually-based activities that allow them to understand the related concept BEFORE we teach the procedure.

These conceptually-based activities COULD be hands-on. But for the above reasons, they can be any activity (a worksheet, an interactive file - preferably requiring student discussion) that offers no procedure for students to follow.

Below are three examples of conceptually based activities that offer no procedure to follow and that require minimal hands-on equipment.

1. Using?GeoGebra?files for students to understand the relationship between the angle sum of a triangle and a straight line.

For those who don’t know, GeoGebra is free mathematics software, which helps engage students with particular mathematics concepts by allowing them to engage dynamically and mathematically. The gif above was created from a file created by Joanne Overs, a past participant of the?GeoGebra-Proficiency course . As you can see, the file allows students to interact with the angle sum of a triangle. Project and manipulate the file on a whiteboard, and the angle sum of a triangle relationship becomes much more evident to students.

2. Showing why:

Quickly show the same pattern for powers of other simple cases, such as 2 and 3, and then ask them for the rule for negative indices. Let the students determine the rule.

Of course, this is not a proof of negative indices, but it doesn’t need to be. We are simply showing students a pattern so they can discover the rule—that way, the rule will make sense.

3. Using the workshopping of 'Open-Ended Questions' to present perimeter and area.

When introducing perimeter to students, the traditional approach is to explain the concept of perimeter and then have students answer perimeter questions.?

Two significant problems accompany this traditional approach:

• Answering textbook perimeter questions is hellishly dull for students.?
• Traditional single-answer perimeter questions do not allow students to understand what they are doing mathematically. We know this because when students progress to area questions (of squares and rectangles), many confuse the routines.

A superior approach is to workshop Open-Ended Questions specific to the related concept. For example:

• Explain perimeter and have students answer one or two traditional (closed) questions.
• Write an Open-Ended Question on the board, e.g. ‘sketch (not to scale) 5 different rectangles, each with a perimeter of 24cm’.
• After a minute (or less if there is a high level of confusion), stop the class, ask for students' solutions and workshop them without letting on whether any particular solution is correct or not. This is key because some students will initially be lost, but once they see some rectangles workshopped, aha moments occur.
• As students progress, some find the task too easy. Challenge those students by adding restrictions, such as:

- Make the total perimeter a decimal number.

- Make your shape a composite figure.

Advantages of workshopping concept-specific open-ended questions:

• Engages students through understanding (understanding leads directly to engagement).
• Requires students to draw on their understanding rather than being required to remember what the teacher said.
• Enables students to understand the concept of perimeter (the workshopping process achieved this)
• It's an activity that is easy to differentiate.

An unapologetic PD promo:

The Workshopping of Concept-Specific Open-Ended Questions is a 4-hour online course. Link here. https://courses.learnimplementshare.com/courses/workshopping-open-ended-questions

Call to Action

What has been your experience with conceptually-based activities? I'd love to hear them.

If you would like to learn more about presenting mathematics conceptually (whilst still teaching procedures but without lots of hands-on equipment), check out some of the courses we offer?here .