The Blog!Easy to read, feet on the ground, relevant-to-the-classroom articles (with no edu-babble).
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Empowering Mathematics HoDs
(The Latest Articles)
Are you on LinkedIn? Then subscribe to our Newsletter 'Empowering Mathematics HoDs'. Short, thought-provoking, sometimes quirky articles to benefit maths leaders and teachers. A new article every week-ish.
- The interview series with Michelle Ang [3P Learning] (17 short videos) Link
- Common roadblocks Maths HoDs face when attempting to influence their teachers’ teaching. Link
- Dividing by fractions: Why most students are confused … and how to make fraction division make sense. Link
- What does success look like in school mathematics? Link
- Why Compartmentalising Is A Bad Idea When Teaching Mathematics. Link
- Trying to improve your mathematics teachers’ teaching? Then have them explore the prairie! Link
- What Percentage of Your School’s Students Dislike Mathematics? Link
- Be proactive, not reactive in the classroom (extinguish the sparks before they become fires) Link
- Start Seeing Poor Student Behavior As A Cry For Help! Link
- It’s Time To Stop Herding Cats In Our Mathematics Lessons! Link
- When Marking Student Work ... Add Dots, Not Comments. Link
- Our view of mathematical understanding is too vague. This one is more precise. Link
- Let’s stop forcing students to learn mathematics by rote! Link
- Why many students find fractions, decimals and percentages confusing. Link
- It's Time To Stop Herding Cats In Our Mathematics Lessons. Link
- What Student Agency Looks Like And Why We Should Strive To Foster It. Link
- Having difficulty dealing With One Of Your Teachers? Adopt The Fireside Chat. Link
- Let’s stop shooting ourselves in the foot in the mathematics classroom. Link
- Let’s Stop Trying To Make Maths Fun! Link
- To Bring About Pedagogical Change In Your Teachers You Need To Understand The Role Awareness Plays. Link
- Could Assessment Be Less Important Than We Think? Link
- Start Seeing Poor Student Behavior As A Cry For Help! Link
- It's Time To Stop Our Mathematics Students From Playing The Memory Game! Link
- When Marking Student Work ... Add Dots, Not Comments. Link
- 4 Ways To Prevent Your Students From Gaining Agency Over Their Learning. Link
Learn Implement Share Articles
The Issue With School Mathematics
- Too Many Kids Hate Maths Link
- Let's stop forcing our mathematics students to play the memory game! Link
- 70% Of Capable Students Are Failing Mathematics. What Can We do? Link
- What does success look like in school mathematics? Link
- What mathematics is and isn't Link
- The Case For NOT Teaching Procedures In Mathematics Link
The Understanding-first, Procedures-second Approach Explained
- Why Students Need To Understand The Concept Before We Teach Them The Procedure Link
- Why We Need An Understanding-first, Procedures-second Mindset When Teaching Mathematics Link
- The Understanding-first, Procedures-second Approach in Action Link
- Why Compartmentalising Is A Bad Idea When Teaching Mathematics Link
- Could These Four Aspects Of Mathematical Understanding Change The Way We Teach Maths? Link
Conceptual vs Procedural Approaches To Teaching Mathematics
- Procedural vs Conceptual Knowledge in Mathematics Education Link
- Procedural vs Conceptual Knowledge - A Hybrid Approach to Teaching Mathematics Link
- Procedural vs Conceptual Knowledge in Mathematics Education - A Classroom Perspective Link
- Conceptual Mathematics Teaching: 4 Common Misconceptions Held By Teachers Link
Teaching Tips - Short Article Series
- Agency Tip #1: One Of Many Is Correct Link
- Agency Tip #2: Dots, Not Comments Link
- Experienced Teacher Tip #1: Be Adaptive Link
- Metacognition Tip #1: Who knows that they know this? Link
- Mathematics teaching Tip: “Forget the maths … let’s understand the question!” Link
- Student Management Tip: See poor Behaviour For What It is - A Cry For Help Link
Teaching Strategies and Tips
- Let’s Put An End To Spoon-Feeding! Link
- The Best-Ever Mathematics Extension Activity Explained Link
- Promote Peer Teaching Between Your Students Link
- Two Metacognition Strategies To Get Your Students Thinking About Their Learning Link
- Using Diagnostic Interactions to Improve Conceptual Understanding in Mathematics Link
- Two steps to minimising student resistance in the classroom Link
- Off-Task Students! One Simple Way To Deal With Them Link
- Why The QAMA Calculator Is A Game-Changer (It requires an estimation each time!) Link
- Three Ways to Authentically Engage Your Students Link
- Sobering Lessons From Observing Teachers Teach Link
have-a-chat videos
- Belinda Naujok
- (00:00) Recent Changes To Her Pedagogy Link
- (04:55) A Student-centred, Conceptual Approach is Ideal For 'Bottom' Maths classes Link
- (11:10) Supporting Students In Need Through a Student-centred Approach Link
- (15:30) The Use Of Mini-Lessons Is An Essential Aspect To This Approach Link
- (20:00) Contrary To The Misconception, A Student-centred, Conceptual Approach Is No More Time Consuming Than The Traditional Approach! Link
- (20:50) Most Teachers Need A Roadmap In Order To Make This Change Link
- (25:00) The Power of Improved Student Understanding Via This Approach Link
- (27:15) The Effect On Exam Results For These Students Who Don't Do Homework Or Revision Link
- Jeff Mills - Full Video Link
- Jeff Mills - gaining Pedagogical Influence As a HoD Link
- Jeff Mills - The Issue of Relevance Link
- James Kiely - This Thing Called Mathematics Teaching (Full Video) Link
- James Kiely - what The Heck Is Explicit Teaching? Link
- Patrick Sinclair - Using GeoGebra Online / Aspects of Version 6 Link
- Andrew Gray - Remote Learning: A Clever Use Of Google Docs With Chat Collaboration Link
- Susan Griffiths - Reflections on remote teaching Link
- Susan Griffiths - The Doc Cam Link
- Ile Ilijevski - Tranformative teaching with GeoGebra Link
- Ile Ilijevski - Getting parents onside with GeoGebra and Remote Learning Link
- Ile Ilijevski - Understanding Ratio and Fractions with GeoGebra (file demo) Link
- Colin Klupiec - The Basics of Teaching Link
- Colin Klupiec - Lesson Structure Link
- Colin Klupiec - Putting Out Sparks Before They Become Fires Link
- Colin Klupiec - All about teacher control (Part 1 of 2) Link
- Colin Klupiec - All about teacher control (Part 2 of 2) Link
- Colin Klupiec - A substitute teaching story (2-minutes, it's a cracker!) Link
- Brad Fuller - E D I A H! (4-min) A story of vision and hope. (We think it's hilarious!) Link
- Brad Fuller - (4-min) What are students hiring school to do? (The answer is a game-changer) Link
- Brad Fuller - Teacher collaboration in a start-up school Link
GeoGebra-based Articles
- GeoGebra Classic 6 vs GeoGebra Classic 5 Getting started with GeoGebra V6 Link
- A Pythagorean Investigation Using GeoGebra How GeoGebra can open doors to new discoveries Link
- Developing Conceptual Understanding With GeoGebra Link
- Are You Utilising GeoGebra? An example of imparting conceptual understanding Link
- Five Tips For Student-Led GeoGebra Tasks Link
- Let GeoGebra Transform Your Teaching - The case for becoming a proficient GeoGebra user! Link
- Investigating The Calculus Sequence With GeoGebra Link
- Exploring Pi With GeoGebra Link
- Demonstration Of An Exceptional GeoGebra Statistics Investigation file (Video) Link
Remote/Blended/Flipped Learning
- Two Common Teacher Concerns When Flipping Classrooms Link
- Six reasons for teachers to use Screencast-O-Matic Deluxe when creating educational videos Link
- Instructional Math Videos: Three Different Types Link
- 6 Reasons Why You'll Need Online Units To Deliver Work To Students Remotely - And Why Google Docs Might Be The Key Link
- Delivering Work Remotely With Chat Collaboration - The Easiest & Best Remote-Delivery Option? Link
- The Flipped Mastery Classroom Explained Link
- Flipping A Classroom: Some Advantages And Cautions Link
- Video Examples Of Flipped Units Link
- Are Instructional Math Videos Likely To Replace Classroom Teaching? Link
Articles About Professional Development
Latest Podcasts
- (Education Perfect podcast) Fostering Student Agency in Maths - With Richard Andrew and Ben Hilliam Link
Older Podcasts
- The best classroom extension exercise ever Link
- Anyone Can Understand High School Maths. No, Really They Can Link
- Teaching Maths Conceptually - An Example Link
- Replace Block Practice With Interleaving In Maths Lessons Link
- Students need to understand school maths Link
- Conceptual vs procedural approaches to maths teaching - Part 1 Link
- Conceptual vs procedural approaches to maths teaching - Part 2 Link
- Student engagement is critical to learning Link
- Achieve better student engagement and learning Link
Articles With Preview
The Understanding-first Approach To Teaching Mathematics
Why Students Need To Understand The Concept Before We Teach Them The Procedure
What bothered nurse Jody was that she was ‘blind’ when following the procedure because she had no understanding of what she was doing. She was genuinely fearful of, one day, accidentally killing a patient ... Poignantly, at one point during this process, Jody looked at me incredulously, and said “You’re kidding me! That’s it? It’s that simple?” Read more |
Why We Need An Understanding-first, Procedures-second Mindset When Teaching Mathematics
Teachers who walk into their mathematics classrooms with an Understanding-first, Procedures-second mindset realise the importance of procedures and procedural knowledge. However, they have as their main aim to have students understand what they are working on for the majority of lesson time. Read more ... |
The Understanding-first, Procedures-second Approach in Action - Three Examples Having students understand the maths they are working through is critical - because the opposite - having students NOT understand what they are working through for extended periods of time prevents learning, breeds disengagement and fosters the ‘I hate maths’ mantra of which we are all too familiar. Read more ... |
Why Compartmentalising Is A Bad Idea When Teaching Mathematics
... the traditional, compartmentalised approach, by default, makes the learning of maths more difficult than it needs to be. I’m suggesting that, by default, compartmentalising maths has more students spending more time in more lessons NOT understanding the work they have been tasked to complete - compared to a less-compartmentalised approach. Read more ... |
70% Of Capable Students Are Failing Mathematics. What Can We do?
Clearly, those who claim to have 'bombed out' with school mathematics never made it to the higher levels. Rather, they failed to cope with simple topics such as fractions, decimals, percentages, basic algebra, Pythagoras, indices, right-angled trigonometry and pretty much anything with a formula. Read more ... |
The Case For NOT Teaching Procedures In Mathematics
I’m not an advocate for avoiding procedures in maths education. However, some research shines a light on the dangers and limitations of step-by-step teaching. Given that, arguably, the vast majority of school mathematics is still taught using - as a default - step-by-step approaches, it might be high-time to spread the message. Read more |
Why We Need To Use Less Block Practice And More Interleaving When Teaching Mathematics
Interleaving, when applied to the mathematics classroom, prevents students from being able to draw upon a single procedure to answer a block of questions. This is because Interleaved questions are mixed questions - usually questions that are mixed just enough ... Read more ... |
Creating A Need To Learn - A Teaching Strategy That Has Students Super-Engaged in Mathematics
Imagine teaching an equations unit in which students: Beg you for your algebraic method / Choose the questions that they work on / Progress at their own pace / Gain instant and repeated success / Beg (more) you for your algebraic method. Read more ... |
Conceptual Mathematics Teaching: Four Common Misconceptions Held By Teachers
Misconception 1: Conceptually-based teaching is synonymous with hands-on activities. Misconception 2: Conceptually-based teaching is too time-consuming to be realistically implemented. Misconception 3: Conceptually-based teaching is too difficult to manage. Misconception 4 ... Read more ... |
Podcasts
The best classroom extension exercise ever
In this podcast, Colin and I talk about ... well ... the best classroom extension exercise ever! What makes this activity so great? It's highly engaging, highly mathematical and requires zero prep! ... Read more ... |
Anyone Can Understand High School Maths. No, Really They Can
In this podcast, Colin and I talk about why it's important to move away from (what I refer to as) a Procedures-first mindset to an Understanding-first, Procedures-second mindset when teaching mathematics to school students ... Read more ... |
Teaching Maths Conceptually - An Example
In this episode we talk through an example of an approach which uses an Understanding-first, Procedures-second approach; what can also be called a highly-structured, conceptual approach to teaching mathematics ... Read more ... |
Replace Block Practice With Interleaving In Maths Lessons
What is Block Practice? Why is the over-use of Block Practice problematic? What is Interleaving? Why is it a solution to Block Practice and why is Interleaving only part of the answer? ... Read more ... |
Students need to understand school maths
This is the third and final part of an ongoing discussion about why it is important that students understand the activities they undertake in mathematics for the vast amount of lesson time at school. ... Read more ... |
More Podcast-Based Articles
Teaching / Tips
Let’s Put An End To Spoon-Feeding!
One of the reasons self-directed learners tend to be effective learners is that they tend to engage in their learning meta cognitively. They ask questions of themselves and others and usually know what they do and do not understand ... Read more ... |
Four Ways Peer Teaching Can Benefit Your Students
The best way to consolidate your understanding is by teaching someone else ... many teachers apply this principle in their classrooms regularly. Below are the 4 ways I’ve seen peer teaching benefit students. Read more ... |
4 Reasons Why Teachers Fail When Implementing New Teaching Strategies
Sometimes we put a great deal of effort into a new initiative for very little reward. It may be a single strategy or something broader. It is demoralising when the hoped-for turnarounds do not eventuate, especially when the time investment has been high ... Read more ... |
Why We Can't Be Lazy When It Comes To
Building Rapport With Difficult Students After I had survived my first couple of years of teaching, I didn’t have too many issues when it came to building rapport with students. However, changing schools always provided challenges, because you begin again with zero rapport and with zero time-grown reputation. This meant that later in my career, due to moving schools ... Read more ... |
Building Rapport Vs ‘Being a Friend’ With Students - Why We Need To Know The Difference.
All aspects of teaching require a teacher to have some sort of rapport with students. Students will retain more information and perform better in classes when the teacher is engaged and shows love for their job as well as an interest in their students. Read more ... |
The Best-Ever Mathematics Extension Activity Explained. (Help - some students are finishing early!)
This is an exceptional extension activity because it 1) is highly engaging 2) is very mathematical 3) has the potential to involve all students, even those not directly undertaking the activity 4) requires minimal preparation. Read more ... |
Promote Peer Teaching Between Your Students
If you are a teacher who understands what it is to be a facilitator of learning then you will know that peer teaching between students is something will occur spontaneously if you allow it to happen ... Read more ... |
Two Approaches Used By Students to Incorrectly Choose Formulas, and Two Things We Can Do About It
When students come across an unfamiliar mathematics problem, they often find themselves at a crossroad when choosing the right formula to solve it. This leads to the adoption of several problematic approaches by students when choosing formulas to solve problems. Read more ... |
Two Metacognition Strategies To Get Your Students Thinking About Their Learning
If we want students to be successful learners - i.e. to take responsibility for their learning - then they need to be reflecting on their learning, on what they know and what they don’t know. Below are two metacognition strategies to help your students Read more ... |
Using Diagnostic Interactions to Improve Conceptual Understanding in Mathematics
It begins with praising all honest verbal answers - even those that are incorrect! Students often give us important, specific information about their current understanding through the verbal responses they give to our questions. This is one of the reasons why it is important to use strategies which elicit verbal responses from students. Read more ... |
Two steps to minimising student resistance in the classroom
You decide to introduce a significant change with your students. But you know you will receive some push back from some if not many of your students. You expect comments such as: “Oh this is stupid”, “This is not fair”, “Why do we have to do this - we liked things as they were before”. Sound familiar? Read more ... |
Off-Task Students! One Simple Way To Deal With Them
Your class is working on a task, but the classroom noise level is creeping up - again - because some students are off-task. You have addressed this situation twice already - each time the class settles for a few minutes before escalating again .... Read more ... |
Why The QAMA Calculator Is A Game-Changer (It requires an estimation each time!)
Estimation is also a skill that should be employed when using calculators. What if there was a calculator that required an estimation before the correct answer was delivered? Well, there is! It is called the QAMA calculator. Read more ...
Estimation is also a skill that should be employed when using calculators. What if there was a calculator that required an estimation before the correct answer was delivered? Well, there is! It is called the QAMA calculator. Read more ...
More Articles About Teaching Tips
Articles About Professional Development
Nine Reasons Why Teachers Are Choosing Online PD Over Face-To-Face PD
A 2009 study concluded that teachers who have attended professional development courses both online and in-person, have ‘preferred to participate online than participate in face-to-face PD. Read more ... |
Five Reasons why Team-Based PD is Ideal for Department-Wide Change
Encourages collaboration and discussion within a Dept ... Collegial motivation to trial new strategies ... Allows Heads of Departments to become ‘pedagogical influencers’ ... Efficient way to bring about department-wide change. Read more ... |
School Leaders: How To Gain Pedagogical Influence Over Your Teachers
During my stint as a Dept Head, the number one thing I wanted to achieve was to have pedagogical influence over my staff. Not that I wanted to dictate the way they should approach teaching but to establish a culture in which .... Read more ... |
Procedural vs Conceptual Approaches to Mathematics Teaching
Procedural vs Conceptual Knowledge - A Hybrid Approach to Teaching Mathematics
There has been a long-running debate about whether a conceptual approach is better for teaching mathematics or whether teaching procedurally is superior. The debate divides mathematics educators into two .... Read more ... |
Procedural vs Conceptual Knowledge in Mathematics Education
Preparing to embrace a conceptual approach to teaching mathematics, including 5 reasons in favour of a procedural (rules-based) approach and 2 reasons why changing from a rules-based/procedural approach can be difficult ... Read more ... |
Procedural vs Conceptual Knowledge in Mathematics Education - A Classroom Perspective
This was me! That students would understand what they were learning was my secondary intention. I hoped students would gain understanding once they had sufficiently practiced the procedures ... I had more than a sneaky suspicion a much better way of presenting mathematics existed, but I also knew I had absolutely no idea what that was. I didn’t know how to make my primary focus the generation of aha moments. Read more ... |
Three Conceptually-Based Maths Activities that Don’t Require Hands-on Materials
Educators often assume maths activities aimed at conceptual understanding to be synonymous with lots of hands-on equipment. However, this is not the case. This article contains three examples of conceptually-based activities that require minimal hands-on equipment Read more ... |
Three Ways Teaching Conceptually Can Save You Time in the Maths Classroom
Many teachers assume a conceptually-based approach is too time consuming to use. However. a well-run, highly structured conceptually-based approach will actually save you time in the classroom .... Read more ... |
Why the Procedural vs Conceptual (Maths Teaching) Debate is Flawed
This article aims to explain why the mutually exclusive ‘Procedural vs Conceptual’ debate is flawed. The argument is made that both approaches are required for students to be successful and that a hybrid approach is superior ... Read more ... |
GeoGebra-Based Articles
GeoGebra Classic 6 vs GeoGebra Classic 5 Getting started with GeoGebra V6
The number #1 area in which Classic 6 is lacking is Dynamic Text. Both Classic 5 and Classic 6 versions have their advantages and disadvantages but it is in the area of Dynamic Text that Classic 6 falls over ... Read more ... |
Developing Conceptual Understanding With GeoGebra
There is a world of difference between students being able to ‘do mathematics’ and students understanding the mathematics they are “doing”. It is clearly advantageous for learners to be both proficient in mathematics as well as to Read more ... |
Are You Utilising GeoGebra? An example of using a GeoGebra file to impart conceptual understanding
To support the argument (that GeoGebra ought to be a key tool in every math teacher's toolkit) take a 30-second look at the dynamic image (gif) at the top of this article. It features a very simple GeoGebra file designed to help convey ... Read more ... |
A Pythagorean Investigation Using GeoGebra How GeoGebra can open doors to new discoveries
One day, I was playing around with GeoGebra and wanted to create a file which visually demonstrated the workings of Pythagoras’ Theorem. I had junior high school students in mind. I wanted the file to cause students to respond with “OK, now I see that ... Read more ... |
Five Tips For Student-Led GeoGebra Tasks
The 'Plug-in-and-demonstrate-with-GeoGbera' method is the one opportunity a math teacher has to make an immediate and significant impact on the engagement of and understanding by his/her students without having to make wholesale pedagogical changes. Read more ... |
Let GeoGebra Transform Your Math Teaching The case for becoming a proficient, inspired GeoGebra user!
Ask a random sample of 50 people between the ages of 20 and 30 how well they understood mathematics at school and I think you’ll agree that ... This article is concerned with the easiest of the three methods, namely plug-in-a-data-projector-and-show-the-file ... Read more ... |
More Articles About GeoGebra
Blended/Flipped/Remote Learning
Two Common Teacher Concerns When Flipping Classrooms
We need to expect some students will struggle with a flipped mastery unit. The transition from dependent learner to independent learner may take some students several months, or more. Guiding students into becoming ... Read more ... |
Six reasons for teachers to use Screencast-O-Matic Deluxe when creating educational videos
Screencast-O-Matic Deluxe will allow you to add explainer text as well as the occasional arrow, highlight and border. You can even blur for email addresses when demonstrating an aspect of an online program ... Read more ... |
Instructional Math Videos: Three Different Types
Any video that attempts to impart conceptual understanding (as opposed to simply teaching a routine by rote method) can be considered a conceptually–based video. Designing a video (or a whiteboard demonstration Read more ... |
6 Reasons Why You'll Need Online Units To Deliver Work To Students Remotely - And Why Google Docs Might Be The Key
Two of the reasons are: Online units make the task of delivering work and instructions to students much, much easier (work can be added and updated at any time whereas a live stream is, well, live!) As a result, the task of delivering a live stream much becomes more efficient because you can reference the information in the online unit rather than demonstrating each aspect live. Read more ... |
Delivering Work Remotely With Chat Collaboration - Could This Be The Easiest & Best Option Available?
This article contains an interview with a teacher who is using Google Docs very cleverly for delivering work to students. Prior to the interview I had assumed OneNote to be the obvious non-LMS to use when creating online work for students. However, seeing the way Andrew uses Google Docs for this purpose has changed my perspective. You be the judge! Read more ... |
More Articles About Blended/FlippedLearning
About the authorHi, Richard here.
There have only been two types of education-based articles that I've ever found interesting and useful. The first type - articles that lure me into thinking about things in new ways. The second type - articles that are highly relevant to my classroom, articles that cause me to say "Wow, I could use that tomorrow". Oh, and they need to be easy-to-read for me even to consider reading them. |
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This is why I strive to write thought-provoking articles that are classroom-relevant, useful and deliberately written using 'feet-on-the-ground' plain-speak.
One over-arching theme meanders through all articles - the authentic engagement of students. Call it my soapbox if you like, but the fact remains that unless students are authentically engaged in lessons, they won't learn anything of value.
Let us know if you think we are achieving our aim and feel free to suggest themes you'd like us to write about.
One last thing before you go - due to the diverse readership, articles are written with a mix of US and UK English, and not without a few 'Australianisms' thrown in. For example, the words math, maths, mathematics, math/s all get a berth. It's not the only broken convention on the site!
Enjoy!
Richard Andrew
One over-arching theme meanders through all articles - the authentic engagement of students. Call it my soapbox if you like, but the fact remains that unless students are authentically engaged in lessons, they won't learn anything of value.
Let us know if you think we are achieving our aim and feel free to suggest themes you'd like us to write about.
One last thing before you go - due to the diverse readership, articles are written with a mix of US and UK English, and not without a few 'Australianisms' thrown in. For example, the words math, maths, mathematics, math/s all get a berth. It's not the only broken convention on the site!
Enjoy!
Richard Andrew
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Understanding & Agency in Mathematics
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