Mastery: Using Maths Talks to develop fluency and flexibility
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By Ian Davies, Mathematics Mastery Director of Secondary.
Do the Maths
Happy new school year! Let’s start with a calculation: think of a few different ways to mentally work out the answer to 13 × 12. Some possible approaches are:
- I know 12 × 12 = 144, so 13 × 12 is 12 more. 144 + 12 = 156. You could call this “using known facts”.
- 13 × 12 = (10 × 12) + (3 × 12) = 120 + 36 = 156. You could call this “using partitioning”, but it also involves “using known facts”.
- 13 × 12 = (13 × 10) + (13 × 2) = 130 + 26 = 156, which is a variant of (2), although perhaps (13 × 10) and (13 × 2) are more likely to be worked out/derived than known.
There are of course many other ways and a great way of starting a lesson is to share strategies like these and to compare and contrast different approaches. As well as possibly learning a few new strategies for future use, students get the chance to demonstrate their reasoning, explain and structure a solution which helps build the skills they need to develop more complex arguments and justifications later. I’ve written before about the meaning of mastery and this is great example of what it means to “know your tables”; being able to recite 1 × 1 up to 12 × 12 is very useful, but if you don’t have strategies to then easily find 13 × 12 you can hardly claim to have “mastered” them!
Equally interesting is looking at errors that might arise here; some students may think 13 × 12 = (10 × 10) + (3 × 2) and there are many ways to illustrate why this is not the case using concrete manipulatives or pictorial representations e.g.
“Maths talks” – in particular here “number talks” – can then be extended as far as appropriate to continue to work on number sense and/or to practice and develop mathematical vocabulary. We place a great emphasis on “talk tasks” like these in the materials we share with partner schools in the Mathematics Mastery partnership. I might continue the above discussion by asking “Now we know 13 × 12 is equal to 156, how can we find 13 × 24?” Double 156 (and there are quite a few ways doing that!) is, to me at least, the most obvious and leads to my next question “Why does that work?”
Well, 13 × 24 = 13 × (12 × 2) = (13 × 12) × 2 using both factorising and the associative law and I think it’s important to use and revisit these terms often so that later on (in the same lesson perhaps) when looking at a more complex examples students understand what they are doing. I’ve heard “explanations” for 3ab × 6a2b = 18a3b3 along the lines of you “do the numbers, then do the as and then the bs”. Compare that to because we’re 3 × a × b × 6 ×a2 × b = 3 × 6 × a × a2 × b × b using the associative law, like we do for numbers and consider which students will have a deeper conceptual understanding and be able to simplify similar expressions in a few weeks’ time?
Incidentally, did you notice that all the methods I suggested for 13 × 12 were examples of the distributive law?
13 × 12 = (12 + 1) × 12
13 × 12 = (10 + 3) × 12
13 × 12 = 13 × (10 + 2)
…maybe a bit of cheeky commutativity thrown in for the last one too. In my experience, students love using the “posh words” and, reinforced often enough, don’t have a problem learning them.
There’s a lot more to say about talk and how to develop productive talk in the maths classroom and I’ll return to this topic with some more examples and ideas very soon – if you have any ideas to share as well, please let us know. In the meantime, all best wishes to you all for the new academic year.
ESPL Program: Dances of the Planets
Venus orbits the Sun 13 times for every 8 Earth orbits. If you track the relative positions of Earth and Venus over an 8 year period, this is the resulting pattern.
The planets in the heavens move in exquisite orbital patterns, dancing to the Music of the Cosmos. There is more mathematical and geometric harmony than we realize. The idea for this article is from a book Larry Pesavento shared with me. The book, ‘A Little Book of Coincidence‘ by John Martineau, illustrates the orbital patterns and several of their geometrical relationships. .
Take the orbits of any two planets and draw a line between the two planet positions every few days. Because the inner planet orbits faster than the outer planet, interesting patterns evolve. Each planetary pairing has its own unique dance rhythm. For example, the Earth-Venus dance returns to the original starting position after eight Earth years. Eight Earth years equals thirteen Venus years. Note that 8 and 13 are members of the Fibonacci number series.
- Earth: 8 years * 365.256 days/year = 2,922.05 days
- Venus: 13 years * 224.701 days/year = 2,921.11 days (ie. 99.9%)
Watching the Earth-Venus dance for eight years creates this beautiful five-petal flower with the Sun at the center. (5 is another Fibonacci number.)
JohnK Wright, V M.S.
Math Teacher at The Einstein School
Technology Resources is a great tool, but not the end all of teaching & education. You can give a student a pencil, paper and a dictionary and they will have all the tools they need to write a paper.
A few years ago, my former school decided to purchase laptops for the school so we had 40 laptops assigned to each classroom. It guaranteed that every student had access to their online high school curriculum, ability to write their essays, research papers, reports, etc, look up information online, etc. What we found out that compared to the previous year where we had about 15 computers in the classroom, and we had to rotate students on and off the computers, while allowing those to work on their bookwork, that our overall school/classroom productivity actually decreased. Why?
1. In the previous setup the computers were facing the wall where the teachers in the classroom could physically monitor what each student was working on. The students not on the computers were doing their work at their tables using textbooks and/or worksheets/reading/etc. This was easily monitored.
2. When we provided laptops to every student, we found that students were abusing the intent of what the laptops were really designed to do as an educational tool. Students were using them to go on Facebook, Chatting, Listening to music, watching Youtube Videos, Watching movies on Netflix,etc. Without having necessary software control to restrict those sites, students were being heavily distracted. We saw our overall Individual Semester Quarter Completions drop as to previous years. Teachers spent more time having to monitor what the students were using the laptops for. Students could easily minimize the windows to “appear” to be working.
3. I still had problems with students not utilizing the potential of what they had. In the old days, one had to go to a library, look for books, get your resources and references. Students had to use problem solving skills to find their solutions. Having access to this technology made it too easy, and with our generation of students who want instant access, this is what we got. Students would type in their math equations, social studies questions, etc, into search engines like Google, Ask, Yahoo, etc. and websites and references would come up with the answers. Students did not have to worry about the process of how they got the answer. That was what was important. I had a student a few years ago who had to write a paper on a famous individual. I read his paper and then I copied some of his text into a search engine. It turns out he copied the entire text from Wikipedia word for word. I failed him and had him redo it. This was a Senior and he was mad at me because I was the first teacher who caught him doing this. He was the same student who would tell me after 5 mins that he could not find a particular topic on the internet and gave up. I typed in some specific words and found several articles that he needed. Kids don’t realize that when they search for information, that there are sometimes hundreds and thousands of other entries not displayed on the screen. It is not feasible or practical to go through that, but this is where students need to be better problem solvers.
4. Just because this generation of students can program applications, use their smart phones, use Facebook, Twitter,YouTube,etc., does not make them technological savvy. Have the same students write a resume, a research paper, essay,college application,etc. The problem is that the technology needs to be treated as a tool. Calculators are useful tools when placed in the hands of someone who has demonstrated the mathematical concepts prior. Laptops and Tablets are great tools for using them to research topics, read ebooks, write essays and documents, prepare PowerPoint Presentations, etc, but it is not just simply turning in your textbooks and giving them laptops/tablets.
Learning is ongoing.
Microsoft Mathematics provides a graphing calculator that plots in 2D and 3D, step-by-step equation solving, and useful tools to help students with math and science studies.