Creating and Exploring Mathematics: How many different directions can we take a topic in 15 minutes?

After reading Bryan Meyer’s thesis (which I highly recommend you take the time to look at), I decided to do something I’m ashamed to admit I rarely do – actually that might be a lie – I don’t think I’ve ever done what I just did. Before I get into it, here’s a quick bit of context…

Bryan Meyer mentions an interesting lesson/sequence of lessons in which he asks the students to explore consecutive numbers in anyway they feel like exploring them. As you can probably imagine, some of the students were completely stumped – they had no idea what he was asking them to do. The students were simply too accustomed to being told exactly what to do and when to do it and had no sense of how to create “mathematical artwork” and play around with mathematics. Essentially, they didn’t know how to create their own mathematics – only how to learn some of the vast array of content which other people have created and possibly apply some of that knowledge to problems given to them by the teacher.

In a Mathemtician’s Lament, Paul Lockhart thinks this is a travesty. He asks the important question, “Why don’t we want our students to learn to do mathematics?”

I could get into a discussion here about external assessment issues, pressure on teachers or teacher perceptions of learning mathematics. Instead, I decided to ask myself the following question:

When have I ever sat down and explored an area of mathematics simply for fun?

I tell my students that mathematics is fun, creative and playful but then I rarely practice what I preach. Instead, I spend time learning about mathematics which other people have created and occasionally add to that by filling in some of the gaps myself – I rarely just mess around with a random topic for fun. Please don’t take this post as confirmation that I’m all about students solely creating their own mathematics and never learning anything that other people have created. I believe in a very balanced approach to a curriculum in which a particular teaching and learning approach during one activity isn’t the best way to do it for another activity. Direct instruction may work for one thing but then exploring a concept using an applet may be better for another. It just depends on the topic and the students you’re working with.

Anyhow, I felt is was highly hypocritical of me to preach the creativity of mathematics and rarely explore things for myself. Hence, I sat down at the table this morning  and purposefully explored consecutive numbers for no other reason than to see what I could come up with and what direction it took me. I didn’t stop when I’d come up with something, I just asked another question which took me in a different direction. After 15 minutes, I was quite amazed at where I’d got to and what I was thinking about.

I stopped at the 15 minute mark and immediately wondered where other people would be at this point – what wonderful mathematics would come out of a discussion between a room of teachers who sat by themselves and explored consecutive numbers for 15 minutes? What would we learn from each other?

It’d be bloody brilliant if there are any teachers out there prepared to spend 15 minutes thinking about consecutive numbers to see what they’ve come up with and where they’ve got to. If anyone does this, I’d be very interested to discuss it. I’m giving a talk next week to our Excellence Challenge students titled,  “Mathematics: Why some people fall in love with it.” It’d be great to discuss the playful nature of mathematics and use this activity as a springboard to expose the students to the true nature of mathematics.

Update: Here’s my 15 minutes of play.  It’s a bit tidier than my original sheets but had to re-write with bigger handwriting so that the photos came out okay. I was pleased to re-discover the integral test for divergence of the harmonic series. I remember learning this about 10 years ago and not having the foggiest where it came from – great to actually discover it for yourself. Infact, when I checked whether my reasoning was correct – I found the exact same reasoning on wikipedia! Outcome: Must do 15 minutes of mathematical play each week.



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14 Responses to Creating and Exploring Mathematics: How many different directions can we take a topic in 15 minutes?

  1. OK – here’s my 15 minutes:

    Looking at the sum of n consecutive numbers – first number a. We have a sequence 2a +1 for 2 numbers, 3a+3 for 3, 4a+6, 5a+10 etc etc – so we can represent this by an + (triangular numbers)

    sum = an + ((n-1)^2 + (n-1)) / 2

    sum = 0.5n^2 + n(-0.5 +a)

    which is the same formula we could have arrived at by using the sum of an arithmetic (with difference 1) ie. n/2 (2a + (n-1) ) = 0.5n^2 + an -n/2

    and so we can have a formula for the average of n consecutive numbers by dividing by n:

    average = 0.5n + a – 0.5

    and then looking at how we define “consecutive” – if we rotate the real number line around the origin and keep a distance of 1 are those numbers still consecutive? eg. rotate pi/3 anti-clockwise and then have all points of distance 1 starting from 0 –

    ie 0, 0.5 + root 3 /2 i , 1 +root 3 i etc etc

    or 0, e^i pi/3, 2e^pi/3

    so we can still find the sum of this sequence using the same arithmetic formula where the common difference is e^i pi/3:

    n^2 /2 e^i pi/3 – n/2 e^ i pi/3

    Further investigation – do these rotated consecutive numbers share the same properties as “normal” consecutive numbers?

    Was quick, so there may be errors in there!

    Interesting idea – thanks.

    • Hey Andrew, great stuff! Would’ve never thought to look at consecutive complex numbers rotated through a certain angle from the horizontal. It’s interesting that you defined consecutive in this sense as being a distance of one away from the previous and the next term. I think if I’d looked at complex numbers, I may have considered “consecutive” to be something like 0,1+i,2+2i,3+3i,… which of course are a distance of root(2) apart. I’ve never thought about what it would mean to have complex consecutive numbers!

  2. Are there numbers that can be expressed as both a sum of 3 consecutive integers AND of 4 consecutive integers? A sum of 4 consecutive integers AND of 5 consecutive integers? Broaden it: A sum of m consecutive integers AND of n consecutive integers? (This was more the result of 15 seconds, not 15 minutes, but those are the questions I would start thinking about…)

    • Hey Matt,

      I’m amazed that those are the questions which initially popped into your head! Some really interesting questions there that wouldn’t have occurred to me at all. Some initial thoughts…

      If you had the same starting number then for your first two questions, one of the integers would have to be zero (that is if we’re just considering consecutive as adding one every time). Infact, you’d have to start at -3 for the first questions and -4 for the second question, and so on. For your broader question, as an example with say 3 consecutive numbers and 6 consecutive numbers – starting at -4 would work (-4,-3,-2,-1,0,1 and -4,-3,-2) but for say 3 consecutive and 5 consecutive, the equations suggest starting at -3.5 which doesn’t work. However, Some great questions to investigate deeply.

  3. Simon Gregg says:

    I read – and was challenged by – Paul Lockhart’s Lament too. (Now reading his Measurement, which is taking me a lot longer as there are hurdles to jump as you read!)

    Ideally students will have had a good solid base of this kind of thing from primary / elementary school. Literally solid. As in, some kind of manipulative.

    I’m thinking about this a lot this year and trying to make sure I use the “classic” manipulatives more. I think it’s partly because they hand more of the control, more of the invention over to students.

    Cuisenaire rods are a good example here – and one that I’ve been reevaluating recently – because they are consecutive numbers and they let your fingers and eyes help your brain do the work.

    See some work – not just on consecutive numbers I’m afraid – from my Year 5 class last week:

  4. Hi Simon,

    It’s funny – I’ve been thinking exactly the same thing as you – trying to get concrete, visual manipulatives to solidify concepts is something I want to improve upon.

    I enjoyed reading your blog post btw – always great to see students analysing sequences geometrically!

    Thanks, Dan

  5. Hi Dan!

    Here’s a record of my fifteen minutes of play. 🙂

    Incidentally, investigating sums of consecutive numbers—as a part of an application to a summer math program—was a key event in my mathematical life. It’s something I’ll be sharing more about in my smOOC.

    Thanks for the prompt!

    • Hey Justin, I’m loving what you’ve done there – especially like the second section on summing alternate postive and negative consecutives starting at one – nice little derivation there.

      I’ve seen a slightly simpler derivation for it. With 1-2+3-4+5-6+7 if you look at it like (1-2)+(3-4)+(5-6)+7 then it’s -1 + -1 + -1 + 7 = 4. Using algebra -1 x (k-1) + (2k -1) = k

      I don’t know why but I didn’t even think of taking the product of consecutive numbers??? Great stuff!

      Thanks for sharing

      • Cool. Even more briefly (or less algebra-y) might be:
        (-0+1) + (-2+3) + … + (-(2k-2)+2k-1) = k pairs that equal 1 = k.
        Also, given previous comments about visualizations, I’ll note what a cool leapfroggy image is generated when you trace out the partial sums of that series on a number line. I think it adds something.
        And I’m looking forward to seeing what turns up in my further investigations of consecutive products.

  6. Ha, that is nice! I like your terminology there as well (leap froggy)! 🙂

  7. Just tried this with my year 10 top set – actually they were a lot better than my year 12s – so maybe students go backwards the more we teach them! These are the edited highlights, which were quite impressive (though the majority of the class did struggle):
    1) What is the greatest number of primes in a set of 10 consecutive numbers? Hypothesis – 2,3,5,7,11 cannot be matched – how can we prove this?
    2) There is only 1 example of 3 consecutive odd numbers all being primes – 3,5,7.
    3) Adding 6 consecutive numbers always gives an odd number. (6n+15) 6n is always even, odd + even = odd
    4) Taking the sequence 1x^2, 2x^3, 3x^4 if we differentiate these functions we get the general formula (n^2 + n) x^n
    5) Does the sum of the fractions with 1 on the top and consecutive prime numbers on the bottom give a sum less than 1? Answer no.
    I particularly liked the prime investigations – looking at the distribution of primes is a really interesting one.

  8. Great questions there and interesting that your year 10 students were more willing to openly investigate than your year 12 students!

    The first one caught my eye as a really nice, simple questions to prove. I had to sit down at break time and quickly prove it. Great stuff! Could have a nice discussion about twin primes here and discuss the recent break through towards the greatest distance between any two primes (which I think has been brought down from 70,000,000 to 4,000,000 now???).

  9. Nice idea isn’t it – and actually links in with current research into the distribution of prime numbers….I think i’ll google some stuff on twin primes for tomorrow’s lesson….

    My rough proof:
    10 consecutive numbers – starting from n with n = odd (we would only start from n = even for 2)
    gives us n, n+2, n+4, n+6, n+8. So this is the maximum number of primes. However any span of 5 consecutive odd numbers must contain at least 1 multiple of 3 (and would contain 2 multiples if the prime number 3 was the starting value) therefore there can’t be more than 4 primes in any span of 10 numbers after 2,3,5,7,11

    it could be tightened up (a little) by using n must be = 0 or 1 or 2 (mod 3) and then subbing in to show that some terms will always have a factor of 3…..

    • Good stuff! Mine of course is pretty much the same – there is always one or two odd numbers which are divisible by 3 within 10 consecutive (since there is a distance of 6 units between odd multiples of 3). Hence max of 4 numbers not divisible by 2 or 3 within 10 consecutive etc. etc.

      So much to investigate with consec numbers!

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