From Understanding to True Application in Math

(This is somewhat a continuation of my previous post titled “Getting Students to Think and Understand Mathematically”)

As teachers, how do we get students to show their true understanding of a mathematical concept without just solving a math problem or explaining the steps? I spend a lot of time pondering this question and would love to get to this point, on a daily basis, in my classroom. What a shift in paradigm that would be … imagine it … a math class that didn’t focus on just skills and solving paper/pencil math problems! One, that instead, requires students to think and stretch their minds and produce work that is connected to real life.

At the start of my “Statistics and Graphing” unit in grade 6 math this year, I asked students to identify similarities and differences in a variety of graphs that they had found. They shared their findings with the class and actually described the different graphs much better than I would have imagined. Immediately I knew that the entire introduction to the unit was a powerful one because the students had done the thinking, connecting, and verbalizing without much guidance from me. This inspired me to continue along this same path … so the next day the students came into class, they saw six different questions on the board, ranging from “How many siblings do you have?” to “What’s your favorite pizza?” to “What month were you born?”. They were set up so that the data collection was done through tally marks.

As we began to enter the data into a spreadsheet, the students needed to figure out the best way to record it. Most students chose a t-table demonstrating the frequency as a whole number. A few students actually figured out how to enter the data as tally marks (I am still not sure how they did that :) ). Once again, I didn’t guide them much other than to explain what we use spreadsheets for and what purpose a spreadsheet cell serves. I wandered around the room for the next 20 or so minutes listening to the little fingers tapping the keys of the laptops, and an occasional student asking someone else for help. Once all the data was entered, we were ready to move onto the application (‘fun’) part of the unit.

Students would come in each day with a new task written on the board. For example, they might see these questions:

Guiding Questions:
1) Look at the different data sets. Which ones could be used to create a simple bar graph? Why?
2) Which ones could be used to create a circle graph? Why?

Students would be required to answer these questions on their own, based on the intro lesson and class discussions that had led up to this point. Once they developed an answer and the mathematical reasoning to support it, we would do the task at hand. Of course I needed to teach the students how to graph using ‘Numbers’ since this entire program was new (and I also created an instructional video for those students who needed to view it a second time) YouTube Preview Image~ but once they got the hang of it, they were able to do most of the graphs on their own. As they would produce the graphs, I would slowly walk around the room and ask open-ended questions like “so what does your graph show the reader?” or “is that the best choice of graph to display the data from your data table? how do you know?”. I would hear students say “Ms. Nave, this doesn’t look right” or “Oops, I didn’t chose the correct graph”. I love the moments when students are able to identify their own mistakes and work through them! Eventually, each student completed the process of making their graphs. Once each graph was completed, they had some “higher-level thinking questions” to respond to based on the graph they completed. (See the attached thought-provoking questions) The unit continued in this manner until all data was graphed and all graphs that the students needed to learn, based on our standards and benchmarks, were made.

We then moved onto the idea of statistics (mean, median, and mode) … and continued to use the laptops to teach and practice these mathematical concepts. We taught our students how to write formulas and sort data within the Numbers program. We provided our students with data that we had collected from the students since the beginning of the year (diagnostic test results, prediction of a minute, MAP test scores). This made it exciting for the students because it was real data! See the attached activities, used within Numbers for Mac.

After weeks of practice, not only in Mac Numbers, but also with stretching their thinking in a different way, the students were ready to show us what they learned and understood about statistics and graphing … so the assessment was created in Numbers. And you guessed it – they completed the test on their computers and submitted it to us digitally. I have included a completed test here: Completed Student Assessment. The true test came about 4 weeks ago when a grade 5 teacher asked if my 6th grade students could teach her students how to graph within Numbers. My first reaction was … yeah, that sounds great … but as I thought it about more, I wondered if I was going to have to reteach everything again. I decided to spend the next math class doing a quick check-in to see what they remembered. I asked my students to open up their laptops and go into Numbers. (Yeah, of course I heard a few grumbles, but who doesn’t when you teach sixth grade?) I posted some make believe test scores on the screen and asked the students to create the best graph to display the data. I would say that about 80% of my students could complete this task individually, and then they were ready and willing to help others in the class. I was definitely pleased with that outcome 5 months after we completed the graphing unit. When they heard that they were going to get to teach this to the 5th graders, they actually cheered!


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One Response to “From Understanding to True Application in Math”


  • Avatar of Nyoli Connor Comment from Nyoli Connor

    Nicely done Allison! I love how you introduced the idea of graphing in such an open way that allowed you to build on student’s current understanding. And as you said, the students brought out vocabulary and classifications with little intervention. Even more impressive is that five months later, your students could still remembered the material. (Now if only I could get those results in my classes.) The idea of using appropriate mathematical language is so important and your tasks addressed this. Looking over the scaffolding of your lesson, the students were able to go over the same material in multiple representations, compare their work with others, reflect and adapt their thinking based on the insights they saw in other groups, communicate mathematically both verbally and in written form creating all the necessary connections! I am impressed. Your students will benefit greatly from the critical thinking skills you are developing in them.


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