Had a nice time in Minneapolis these past two days at the #NCTMRegional. It must get ridiculously cold here in the winter since they built an entire interconnected-Skywalk throughout the city. I think I only went outside twice the entire time I was here – getting in and out of the taxi! (Which was pretty terrific as the first two days were rainy).
I went to a few sessions this time that really got me thinking about the importance of questioning in mathematics. Even when utilizing technology or hands-on manipulatives/resources, the questions we ask the students are vital in order to deepen their understanding and encourage discourse and exploration. Questioning to me is the most important skill a teacher can develop to help their students – more important than any resources or technology that might be available, because it is only through questioning that we foster rigor and develop deeper thinking to help students understand and make connections in mathematics.
What I loved about two sessions in particular that I attended, a 6-12 Statistics session with John Diehl, and a 3-5 Making Math Meaningful session with Jennifer N. Morris, was the focus on questions and how the same mathematical concept could be appropriate for students in any grade depending on the questions asked. In John’s session, where we were looking at bivariate data, we used data athletes, made a scatterplot on the Prizm, and then had great discussions about lines of fit, variables, causation, association and a multitude of other ideas around helping students understand what the data represents, in context, and how, depending on the question asked, you could address algebra content or calculus content. Students in sixth grade can be do linear regression simply by asking the right questions and allowing them to explore their conjectures with technology, such as the Prizm. The questions lead to discussion and exploration and more importantly, to more questions that the students themselves begin to ask. Something as simple as “should your graph go through (0,0)? What does that actual mean in relation to the data and does that make sense?” helps students apply math content to the real-world context and make sense of the data and the graphical representation (sounds very Common Core to me!)
In Jennifer’s session, the first part was creating an Origami pinwheel, which seemed relatively simple but the questioning throughout about area of folds, and fractions of the whole, and how do you know, what shape do you have now and what’s the ratio of this shape to the shape before – really demonstrated how a seemingly simple hands-on activity can be full of rigorous mathematics and mathematical connections. And – the same activity would be appropriate for multiple grades – up through geometry even, simply by changing the questions you ask. When we began working with the Casio fx-55Plus, Jennifer did several quick activities using the Random# generator on the calculator, but the questions asked really had the teachers (i.e. students) thinking about numbers, fractions, comparing numbers, estimation, reasonableness, probability. For example, if these random numbers (all fractions, but with varying numerators and ugly numbers like 651/1000) represented the part of a cookie your mom was going to share with you, when would you consider it big enough and why? This led to really interesting discussions on how to determine
what fraction you had of a whole and is just being more than 1/2 enough. She had participants stand up front with their calculators & their randomly generated fraction and rearrange themselves in numerical order – not so easy when the fractions are 516/896 and 37/52 for example. Seemingly simple activity, using technology to quickly generate numbers and have really rich discussions that help make mathematical connections. And it came down to the questions asked – engaging, in-context, and appropriate for many grade levels.
Both sessions confirmed for me something I have always believed – the questioning is the easiest way to get your students thinking, talking, applying and connecting math. Learn to ask questions and you can create an engaging learning environment that differentiates the learning and provides students with multiple pathways to make connections.