I want my year 11s to put some practice in to learn the circle theorems word-for-word. To make it a bit more interesting for them I’ve put together:
Feel free to use with your students to if you like them
I’ve created a page on Great Maths Teaching Ideas which features interactive versions of all the circle theorems. Go to the ‘shape’ menu and then click ‘circle theorems’. Enjoy!
Alternatively, here’s a link to the page: http://www.greatmathsteachingideas.com/circle-theorems/
Here’s an example of what you can find on the page:
Alex Bellos of Alex’s Adventures in Numberland fame has written an article in The Guardian about a stamp-series in Macau. (Stay with me!) The series highlights the history of Magic Squares. The stamps, worth 1-9 Patacs are even bought on sheets arranged into Magic Squares based not their monetary value! Could your students design a stamp series around a different mathematical topic?
An interesting read: Macau’s magic square stamps just made philately even more nerdy
I’ve put together a revision programme based on MathsWatch for the year 11 cohort of 2015-16 studying Edexcel 1MA0 GCSE maths. It is based on spaced-learning principles to maximise retention. This could be used as an additional revision programme to complement weekly practice papers, which are essential in my opinion. It will ensure students systematically visit all topics on the GCSE 3 separate times during year 11 revision.
1st revision– students watch the MathsWatch full video lesson and make notes on this template.
Also, download an example you can show them of what good notes look like.
2nd & 3rd revision– students watch the MathsWatch ‘1 minute maths’ video and complete the questions in the video.
Little and often! A bit of structure for the well-intentioned students who need it.
*Note to be aware of… Clip 60… We teach Gelosia multiplication at Wyvern College, rather than long multiplication. The pdfs advise students to look for a clip elsewhere on this topic.
Today is a rather emotional day for me. The last few years have been an amazing journey learning about effective maths teaching. Steps along the way have included: The problem with levels- gaps in basic numeracy skills identified by rigorous diagnostic testing, Forgetting is necessary for learning, desirable difficulties and the need to dissociate learning and performance, Going SOLO on the journey towards deep learning,How do we make John Hattie’s “Visible Learning” work in maths? and, of course, You’ve never seen the GCSE Maths curriculum like this before…
Today is a milestone where I’m launching Numeracy Ninjas, a free KS3 numeracy intervention that puts all of my CPD learning from the last 3 years into practice! It incorporates all of the curriculum mapping work with a schedule that maximises retention over time. Best of all, I’m making it completely free for any school to use!
Visit www.NumeracyNinjas.org to meet the Numeracy Ninjas!
Download Powerpoint here.
I have felt for a long time that one of the disadvantages of the current levelling system is that it encourages teachers to constantly teach students mathematical concepts and ideas only at levels equal to or above that which they have recently scored on an assessment. The data-focused, level-centric system only rewards teachers if their students can score marks on higher-level content; there is no explicit incentive for filling gaps in students’ knowledge at lower levels.
Good maths teachers know the importance of their students having a strong knowledge in the foundations of the subject and the long-term benefit of plugging any gaps students have. This year I wanted to be much more systematic in identifying any key gaps students had in their knowledge of the foundations of secondary maths as soon as possible when they arrived in year 7.
What we did
We diagnostically tested every year 7 student in three areas: mental numeracy calculation strategies, timestables and key nodes. For each student, we assessed 30 mental calculation strategies such as: number bonds, reversing an addition sum to make it easier and counting from the smallest number to the largest in subtraction etc. We assessed 30 of the timestables. The top 30 key nodes topics identified from my previous work on understanding the most important topics for students to master prior to studying GCSE maths were also assessed. In total we collected 90 data points for each of 203 year 7 students shortly after they arrived with us in September 2014.
The testing was carried out in a single 50 minute lesson using the QuickKey app available in the App Store. Students were shown a PowerPoint presentation that contained the 90 questions, each with a specified time limit. Teachers click ‘start’ on the presentation and students had to identify the correct answer for each question from five multiple choice answers. We ensured ‘distractor’ answers were placed alongside the correct answer in each question based on common misconceptions. Students each had a grid on a piece of A3 paper in which they had to colour in the correct circle corresponding to their chosen answer for each question. After the assessment the test papers were scanned using the QuickKey app on my iPhone which automatically marked them and recorded the whole cohort’s results in a single spreadsheet file. The scanning took just 3 hours and we obtained over 18,000 data points for the whole cohort from just a 50 minute lesson!
We tested the following number of students in each KS2 sub-level group:
3.0- 4, 3.3- 5, 3.7- 9, 4.0- 23, 4.3- 28, 4.7- 32, 5.0- 30, 5.3- 35, 5.7- 16, 6.0- 21
This graph shows the performance of different KS2 sub-level groups in the three main areas assessed.
The Pearson correlation coefficient for the mental strategies, timestables and key nodes assessments versus KS2 level were 0.72, 0.58 and 0.71 respectively. These correlations were also checked against a KS3 SAT assessment sat by all students during the first half term with us and found to be almost identical. The weakness in the mental calculation strategies of level 3 students is clear to see. Level 4 students were far from being secure in many of the basic mental calculation strategies we would take for granted that they would know. For example, some did not automatically reverse an addition to make it easier, could not calculate a number bond to 100 or tell the time. It is important to state the timing of the questions was reasonably swift during the mental calculation strategies and timestables section of the assessment as we wanted to assess what students could do fluently through recall and fast strategies, rather than what they could do with written calculations if they had a lot of time. During the key nodes assessment we gave students a bit more time and allowed them to use written calculations, but again set the timing such that if they did not show good fluency in choosing and executing the correct strategy they would not have had time to answer the question. So when I claim that some students could not do number bonds to 100, I am saying that they could not do this mentally within approximately 10 seconds. The thinking behind this assessment approach was that in order for these skills not to become barriers to learning and working-memory-consuming difficulties when studying more challenging topics on the GCSE course, we want to assess whether these skills are fluent and can be executed quickly, almost without thinking.
The results of the timestables assessment were better than I expected, particularly for the lower attaining KS2 students. However, when looking into the data it was quite apparent how many of these students who knew their timestables, perhaps did not understand the concepts behind them. For example, they got the questions on understanding multiplication by its link to repeated addition wrong.
To delve a little deeper, the following diagrams show the mean average proportion of students within each KS2 sub-level group that got each of the 90 questions correct.
Mental numeracy calculation strategies
There are many interesting interpretations and observations to be made from these results. I will leave it to you to delve into this as deeply as you wish. I think the diagrams summarise nicely the differences between what level 3, 4, 5 and 6 KS2 to students can do fluently across the topic of number. Broadly speaking: level 3 students have mastery of none of the three areas; level 4 students have some proficiency with mental calculation strategies and timestables, but not the key nodes; level 5 students have reasonably secure mental calculation strategies and timestables and have some proficiency with the key nodes; and level 6 students are broadly secure in mental calculation strategies, timestables and can do many of the key node topics already. It is an obvious statement, but with the strong correlation coefficients already cited, it would appear that what a student can do in number is a strong indicator for what they could do across the curriculum.
These results add further support to my belief about levels hiding gaps in the foundation knowledge of some students. A significant proportion of 4 students struggled to fluently identify number bonds to 100, a level 3 skill. Even many of the level 5 and 6 students did not answer the questions on understanding multiplication as repeated addition or division as the inverse of multiplication correctly. It goes to show- giving a ‘level 6 learning objectives’ sheet to a ‘level 5′ student is not good enough. Perhaps they could already do some level 6 topics, but may have gaps at levels 3 and 4. It must be personalised on a individual student basis.
Has growing up in a levels-based system, where teachers and students are only rewarded for achieving success on content at higher levels in the subject, resulted in oversight/ignorance of the gaps in the students’ foundation knowledge at lower levels? How much easier would learning the more advanced topics be for students if they had comprehensive fluency with these basic skills? The hidden gears need oiling once in a while.
To be clear, this is in no way a reflection on our excellent primary colleagues. They do a brilliant job. They are constrained by a single-minded, level-incentivised, high-stakes system, just like we are in secondary, and they act accordingly to meet the external pressures placed on them. We do the same in Year 11.
A change in mindset is required. If levels are going (and they are!) we must not replace it with a system that has the same flaws. I am certainly not suggesting that we shouldn’t teach students higher level content than they can currently attain; but this must not be just a single-minded focus either. From the diagnostic testing we did this year I have learned that KS3 lessons across the ability spectrum still require systematic, planned, regular practice in building students’ fluency in the foundation topics of number. Fluency (speed and accuracy) is a fitness, it is not binary. Even if you are 100% accurate, you can always be faster then you are at present. If two students can calculate number bonds to 100, but one of them takes five times as long to do it as the other, their learning of higher-level concepts will be all the more difficult for them later in their maths education. We must not fall into the trap of confusing instantaneous performance for retention and transfer – learning. I have written about this extensively before. Learning must not be seen as a checklist of visit-once objectives. Even the highest attaining students need to occassionally revisit some of these elementary topics for which their “fluency fitness” has fallen. A green cell in a spreadsheet indicating that they can do a skill today should not be taken as a proxy that they will be equally as fluent in this skill in six months’ time.
In our KS3 lessons at Wyvern College from September 2015 we will ensure that not only do we strive to raise students’ proficiency with higher-level concepts, but we will also provide short daily exercises that build students’ speed and accuracy in all three areas assessed in our diagnostic trial. It is with great excitement and anticipation that I am going to launch a five-minute-every-lesson, fluency-building product on Great Maths Teaching Ideas this summer. It will provide systematic, rigorous coverage of every topic assessed during our diagnostic trial. Watch out for that! As a consequence of what we learned from the diagnostic testing of our year 7 arrivals this year, I feel so passionately about the need for this product, that it will be made available free for all schools via this website in the next month or two.
Keep an eye out!
Update: the Powerpoint we used can be downloaded here.
Teaching time-series graphs?
Get students to work in pairs. One solves a jigsaw puzzle. The other student records how many pieces in the jigsaw were solved every n seconds. Choose n appropriately.
Students then plot graphs of n.o. pieces solved vs time. Then ask them:
AQA have recently released their ’90 problems’ to support teachers in delivering the problem-solving element of the new 9-1 maths GCSE. It’s a rebranding of the 90 problems they released to support the teaching of Further Maths GCSE in 2008. There a number of issues with many of the diagrams on the newer version, so I’ve attached the 2008 version for your interest, here: AQA 90 Problem solving questions. Answers
They are excellent resources and perfect for some little-and-often problem solving practice. Highly recommended.
A lovely idea from a colleague- regions stained glass windows…
Look at the shapes in this square grid. The hexagons on each row, column and diagonal are made of the three shapes in that row, column or diagonal respectively. This completed ‘Geometric Square’ was posted on Twitter by @PardoeMary.
Can your students create another geometric square? Can they create tessellation patterns of the results? What if you didn’t use hexagons, but some other starting shape?
Click here to download a blank template: Geometric squares
Optimus Education have launched a new conference: English and Maths 2015- Effective Teaching Strategies to Meet New Accountabilities.
I’m delighted to have been asked to speak at the event on 22nd October and will run a workshop session on: designing a maths curriculum that ensures learning is conceptually deep, retainable and context-transferable; maximising depth of learning and procedural fluency.
Other speakers include Matt Parker (of Numberphile and Standup Maths fame), David Didau (influential blogger) and Danielle Bartram (author of the well-known @MissBsResources).
Download a pdf brochure for the conference here.
It would be great to see you there! Book early for up to a £60 discount.
Kahoot is a tremendously useful, free AFL tool I recently came across after a Twitter recommendation. Students can use any web-enabled device (any OS platform) to take part in games-based quizzes. There are thousands of quizzes publicly available, or you can create your own in a simple drag-and-drop editing tool and add to the public pool. There are many maths-based quizzes already in the public pool suitable for KS3 and GCSE maths classes.
Students can access the quiz without needing logins and passwords, making it easy to use in class. They simply go to kahoot.it in their web-browser and type in a game-pin number. The game begins and students answer multiple-choice questions, scoring points for speedy correct answers. The leaderboard is updated after each question to build the tension!
After the quiz you can download a colour-coded spreadsheet of all your students’ responses. The students love it- I love the AFL!
See the video below of Kahoot in action to get a feel for it. To register for an account, go to www.getkahoot.com.
I have become increasingly interested in visual models recently as a way of introducing topics. Visual models have the power to illustrate concepts in their rawest, simplest form without the misleading associations that words and abstract notation can introduce. I’m convinced concrete/visual model introductions should form an increased part of my practice, but the question that interests me is which visual models should I use? There are obviously many considerations, but one is how comprehensively they cover the syllabus. A visual model that demonstrates expanding brackets particularly well would not be that useful if it could not also demonstrate the concept of factorising. Through much reading I believe I have found two visual models that cover the vast majority of number and algebra topics. They are almost mutually exclusive too, complementing each other by covering different, rather than overlapping parts of the curriculum.
The first visual model is bar modelling. The more I experiment with it in lessons, the more I am convinced it can open the door to so many topics that many mid-to-low attaining learners previously found inaccessible. I wrote recently (click here to view) about all the many different topics bar modelling can be used for- from basic fractions work, through FDP, ratio and up to reverse percentages and compound interest.
In contrast to bar modelling, I believe algebra tiles is a very powerful concrete/visual modelling technique that can be used to develop conceptual understanding of topics. In this post I will explain how the algebra tiles model works and demonstrate how it can be used to introduce a great many topics that bar models are not suitable for.
Before I go any further I want to make it clear that I do not think all students need to experience visual models when topics are introduced. Teachers should use them selectively when they think they are needed and will support students’ learning. Visual models often fall down for particular variations of question and they are not meant to replace abstract reasoning, merely be a bridge to it for students that need it.
Basic rules of algebra tiles
It is anticipated that you will make these tiles as physical resources for students to use.
Four operations using algebra tiles including with negative numbers
Types of number
All of the above concepts are relatively trivial and you may have used variations on the algebra tile model when teaching them. For example, using multi-link cubes to derive factors or show the square numbers. However, what is powerful about the algebra tiles model is how seamlessly is transfers to algebraic concepts…
Collecting like terms
Completing the square
Like bar modelling, I like the way that algebra tile models cover so much of the syllabus. Whilst bar models cover mainly FDP and ratio, algebra tile models seem to cover the other topics in number and algebra. Rather than competing with bar modelling, it seems to complement it.
I’ve not used algebra tiles in my lessons yet and will do some experimenting in the coming months. I particularly want to focus on whether their use enhances students’ understanding of concepts- whether they see the links between the visual and the abstract.
At present I’ve not found many resources out there to scaffold and support this approach. If you know of any please do let me know so I’m not reinventing the wheel! If you have a go with algebra tiles (or bar modelling) yourself let me know how you get on and what you learn as you go along.