We talk a lot about how learning math should NOT be focused on memorizing steps and formulas, but instead on doing and discovering the relationships and building understanding. Ideally, students should be visually seeing multiple representations of concepts (tables, graphs, words, pictures, models, manipulatives, equations, etc), comparing and analyzing these representations, and making their own conjectures and ‘rules’. Instead of telling them what they should see or know or do, they are able to figure it out themselves by focusing on relationships and connections (obviously with good structured investigations and questions and activities). This is the ideal.
Research shows that when students discover the patterns and come up with rules/formulas/relationships themselves, they are much more likely to retain the information, or more to the point, more able to ‘recreate’ the experience they had and recall and/or rebuild the information. Not possible when memorizing isolated facts or skills. It’s why just teaching struggling students rules and steps and skills and making them repeat it over and over does not improve learning…..there are no real connections being made.
Algebra is a subject that many students struggle with because of the ‘unknowns’ or variables – it is abstract, and without exploring patterns and relationships of the variables in many forms, it is hard to help students understand and make sense of things like “equations to model real-world situations”. Sadly, there is still the tendency to just teach process – i.e. follow these steps, or “use the quadratic equation to get the solutions” without even talking about what those solutions mean, where they are located, why there are two (or one or none). If we just teach skills and if students just memorize processes/steps, when confronted with something similar, but not exactly the same, they can’t do it, and give up, because they have not had experiences that help them build understanding and see relationships and connections to prior knowledge.
This leads me to today’s focus – Dynamic Graphing.
Dynamic mathematics is when something in a given representation is changing (a measure, a value, a construct), and as it changes, you see the impact of that change. As an example, in geometry, this might mean a triangles three angles are measured, and as you move one or more of the vertices, the angle measures change as you see the triangle change. If you sum the three angle measures, you get 180, and as you change the triangle dynamically, even though the triangle is now different, with different measures that are changing as the triangle changes, the sum of those angles is still 180. Which leads to a conjecture.
Dynamic graphing is the same thing – it’s using variables (instead of static numbers), in equations and changing one or more of them, and then watching what happens to the graph as that variable changes. Dynamic graphing really emphasizes what a variable means – i.e. a quantity that varies, and as it varies, the graph/tables/equations also change. As an example, in a linear equation, defined as y=Ax+b, if we make A- a dynamic variable and change it, students will see the line change it’s ‘steepness’ and direction, and make a conjecture that A- must have something to do with a line’s steepness. If we change b- dynamically, they will see the line move up and down vertically (but not change it’s direction/steepness) – so they conjecture that b- determines where the line crosses the y-axis (because they notice the line always seems to cross at the y-axis at a point that corresponds to b-). Simple things like dynamically changing and visually seeing the impact of that change can lead students to more deeply understand what those coefficients/variables in an equation mean and do, so that when you give them y-=-3x+5, they already know the direction of the line (negative), how steep it might be, and where it’s going to cross the y- axis. You didn’t have to tell them – they figured it out by observing, looking for patterns, and making conjectures and then, most importantly, having discussion with others to confirm their findings.
I wanted to share how you can do dynamic graphing on the new fx-9750GIII graphing calculator. Not a new feature – it’s possible on all the Casio graphing calculators. However, I don’t think many people realize this functionality exists, especially on a black-and-white, inexpensive (but powerful!) graphing calculator. It’s something I think many people expect only mathematics software to be able to do, so I wanted to show you this feature. (And, you can do dynamic geometry too with the Geometry Add-in Menu!)
Be sure to visit Casio Cares: https://www.casioeducation.com/remote-learning
Here are quick links: