In my last blog post, I shared my mental model for how to teach a lesson that incorporates both student-centered inquiry and direct instruction. There are eight steps in this instructional model, but it really is three main phases: introduction, inquiry, and direct instruction. This lesson structure is very similar to the 5 Practices for […]
In earlier blog posts (this one and this one) I’ve talked about two – seemingly opposing – views of math instruction: direct instruction and student-centered instruction. There is plenty of evidence that students benefit tremendously when given an opportunity to invent/investigate their own understanding prior to formal instruction by the teacher. There is also plenty of evidence that […]
The California Mathematics Framework calls on teachers to teach conceptual understanding, procedural fluency, and application with “equal intensity”. The way I think of “equal intensity” is with a Venn diagram. I first saw a Venn used to describe rigor here. The ambitious instruction we are seeking – rigor – is the sweet spot in the […]
When it comes to mathematics instruction, what are we aiming for? Exactly what do we want students to know and understand and be able to do? Do we want students to have a big-picture understanding of the mathematics they are learning? Do we want students to be skillful in using procedures and algorithms? Do we […]
Finding the area of polygons is pretty easy if the shapes are familiar to you like rectangles, triangles, circles, etc. Those shapes have nice formulas we can learn that will give us the area of the figure without much hassle. But what about unusual shapes like this? Sure we could find the area of this […]
Children should be led to make their own investigations and to draw their own inferences. They should be told as little as possible, and induced to discover as much as possible. -David Eugene Smith (1904) I will never forget the day I finally became a mathematician. Was it when I was placed in the […]
A teacher recently asked me to co-plan a lesson with her and then co-teach it. She wanted to teacher scatterplots and found this activity, but she wanted my support in planning and teaching the lesson.: https://teacher.desmos.com/activitybuilder/custom/5ae8aa5436ae340a1d96ad4c We began by printing the Teacher Guide to see if the activity’s author provided any suggestions for pacing and […]
Take a moment to solve this problem using any method you wish… A farm has a total of 480 llamas and alpacas. of the number of llamas at the farm are equal to of the number of alpacas. How many more alpacas than llamas are there? I’ll wait. . . . . . Can I […]
How do your expectations influence the performance of your students? The answer may surprise you, and, like many influential scientific studies, it starts with some dumb rats and some smart rats. Bob Rosenthal, PhD, is an experimental psychologist has spent the last four decades researching how student performance might be influenced by our expectations. In […]
Every three years the OECD administers and publishes the Programme for International Student Assessment, better known as PISA, which evaluates 15 year-old students around the world to determine how well their education system has prepared them for life after compulsory schooling. This test is important because it allows the performance of educational systems to be […]