Two of our authors, Nick Broom and Tim Hixson, break down what each component of the toolkit means, and how you can integrate these tools into your teaching.
What is the mathematical toolkit?
The mathematical toolkit provides students with time to work on activities that reflect real world problems and the tools they need to solve them. The mathematical toolkit has a goal very similar to Kognity’s. The underlying premise of the toolkit is active learning. Instead of listening to a teacher talk about a topic, the toolkit encourages teachers to allow students to engage with the material and develop skills and techniques that they can then add to their own personal “toolkit”.
The 5 parts of the toolkit are:
- Cognitive activators:
These are strategies intentionally used to get the attention of students and to motivate them to explore new concepts at the beginning of a new topic. These activators connect previous learning to new topics and review or introduce new skills.
- Conceptual understanding:The twelve fundamental concepts enable teachers to frame learning in a way that emphasises the core ideas that connect topics throughout the syllabus. Encourage students to connect with the concept underpinning a situation rather than focusing on choosing a formula to use.
- Using technology:Students are encouraged to appreciate the connection between technological and mathematical advancements. They are also trained to utilise technology properly, in the appropriate contexts and with fluent efficiency. These methods enable teachers and students to explore mathematics extending beyond what they have the skills to do by hand, to examine data to identify patterns, and to enhance collaboration and communication about mathematics.
- Mathematical modelling:The main purpose of these modelling activities is not just creating a model, but rather, to follow through with a design process that includes testing the model, reflecting, then extending and improving it. Constructing tables, charts, graphs and a variety of other models to visualise data, algebraic relations and geometric structures enables students to make connections and explore relationships and properties to enhance understanding.
- Proof:Deriving various properties, theorems and formulae in a manner similar to the mathematicians who originally discovered them deepens understanding and enables students to get a glimpse of how knowledge is created and validated in the area of mathematics.
How can I integrate the toolkit into my teaching?
When choosing or creating an activity for my class, I try to choose activities that are open-ended and do not have a clear path to the answer. I often say to my students that I don’t care how they get to the answer as long as they can explain their thinking and back it up with evidence. The activities are not always successful in terms of getting a correct answer, but that is not the point. By working through activities, students can develop an understanding of which techniques work for them and how they can best approach unfamiliar problems. Also, by keeping the activities open-ended, there are more opportunities to discuss the bigger concepts being used.
The Kognity maths texts also have numerous features to help you integrate the toolkit into your teaching.
Each subtopic is introduced using a “The big picture” section that makes connections and demonstrates the need for understanding in each new area. To integrate cognitive activators, you can use the activity, discussion, or video in “The big picture” to give context to the material that the students are about to explore. These sections also aid conceptual understanding as they explicitly link each subtopic to one or more of the fundamental concepts.
Technology use is supported via images and tutorial videos throughout the text that explain how to use key features of four graphic display calculators commonly used in classrooms today. Students can explore how technology can be used to aid their work. Whether by using Microsoft Excel to create a Monte Carlo simulation or by using Desmos to visualise transformations, the goal is to include a broad range of technology to help students understand its power and use. Kognity embraces this by including interactive content throughout the textbook.
Technology and other graphics are used in a variety of ways to illustrate mathematical modelling. Students are also encouraged to explore relationships by manipulating models presented in embedded Geogebra Applets. Finally, proofs of many of the key theorems and formulae are provided, regardless of whether students will be expected to work them out by hand or with the calculator.
In addition to these features, investigations are provided at the end of each subtopic to encourage students to explore each subtopic further. They often include activities that can include this design process for mathematical modelling. Teachers can use these investigations to reinforce all five elements of the toolkit and develop greater understanding and problem solving skills.
Authors: Tim Hixson and Nick Broom. Tim is an experienced mathematics and technology teacher, author and IB examiner. Nick is a senior IB examiner, author and experienced mathematics and physics teacher. They are both based in the US.