It is important to remember that when we engage students in experiences meant to help them build meaning behind math concepts, what might appear to be happening may or may not be actually happening. Let me explain. Place value is one of the times in math when understanding what the concepts represent requires more than students just repeating back the language.
A quick internet search reveals thousands of place value charts that attempt to connect the language of place value with the digits in each place. Some have just the names in columns above each place and others have pictures of commonly used base ten blocks. Students use these charts to say that the 6 in 168 is in the “tens” place and that it is equal to 60, but what have they really constructed about the concept? What do they have to understand?
A student in the FPU teacher education program, Keely McGee, had this question and engaged her class in a lesson on place value that involved counting cottons balls. This was from her unit on local resources and cotton was one crop discussed.
Keely had students answer the question, “How would you count all the cotton in the giant pile shown in the picture?” Like the AIMS composite unit activities described in the series “What Every Student Needs to Know For Multiplication,” she had students build composite units. The difference is that she had them build the cotton balls in groups of ten that would become a basket of cotton balls. Each of these composite units of cotton balls were put together to make a composite of composite units; ten baskets then became a bale of cotton. This concept is more sophisticated that we sometimes believe so this didn’t automatically mean students built an understanding of place value the next day.
After sharing this activity with me in class, we discussed providing students with time and regular opportunities to reflect, and how, by encouraging students to reflect on how this relates to our place value system, the words one, tens, and hundreds could hopefully hold meaning. Keely wanted her students to not just label the places with names, which help only cursory meaning, but she also wanted then to build meaning that would serve them as they perform operations with those numbers, and as they later try to make sense of fractions and multiplication.