by Connie Rivera

In my class, I regularly ask students to:

- Match to each other cut outs of different representations such as: a graph, table, or equation matched to a situation; an array matched to an expression; or a drawing matched to a fraction, decimal or percent.
- Sort into categories shapes, graphs, or different visual and symbolic representations of math concepts.
- Place on a number line, whole numbers, fractions, decimals, and percents as well as representations of these as groups of objects, time, money, and parts of a whole.

I encourage communication between students during these categorization activities by asking them to state their reasoning out loud to a partner or their group before adding or moving any of the pieces. Stepping back, watching, and listening allows me to quickly assess students’ level of understanding as well as spot their misconceptions. After I’ve given students a chance to correct their own and each others’ mistakes, I might ask a few questions:

*Are there any pairs that could be combined because they are equivalent?**Are you happy with… the distance between these? Are there any you’d like to rearrange?*

Here’s the hard part for me – if they still don’t catch their mistakes, I don’t correct them. Most of these activities have pieces that can be taped together and posted to look at later. The number line for example, lends itself to being added to over time as their understanding refines with new lessons. As Jo Boaler describes in *What’s Math Got To Do With It?,* brain research tells us that students have two opportunities to “learn from a mistake.” The first learning opportunity occurs when a student makes the mistake, *whether or not they are aware of it*. The second learning opportunity comes from catching that mistake themselves later on.

These experiences are for students at different levels, working together to learn. A description of a solution can appear to make sense when we hear about it, but turning to explain it to someone else is how we know we really understand. These activities allow our learning to start with a more hands-on, concrete and pictorial exploration before moving to more abstract representations. Through the exploration, students make connections between the visual, symbolic, and quantitative representations a concept. Sorts, matches, and clipping representations to a number line enable all students to use these connections to *make sense *of math.

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*Connie Rivera teaches numeracy skills to adults of various skill levels, including court-involved youth and English Language Learners. She is also a math consultant, providing math strategies and support to programs implementing the College and Career Readiness (CCR) Standards for Adult Education. As a consultant for the SABES numeracy team, Connie facilitates trainings and guides teachers in curriculum development. Connie is President of the Adult Numeracy Network and a LINCS national trainer for math and numeracy.*