# Directed numbers

Discussion in 'Mathematics' started by mon_ster, Oct 6, 2015.

1. ### mon_sterNew commenter

Can anyone tell me if they have found a successful strategy for ensuring pupils can calculate consistently and accurately with directed numbers. I find very few pupils ever obtain complete fluency with this aspect of calculation despite many showing conceptual understanding if they slow down and really think about it.

2. ### coyoteNew commenter

What's a directed number?

3. ### JaquesJaquesLiverotEstablished commenter

The problem a lot of people have is that negative numbers don't "exist" in an obvious way; you can't have -5 apples, for example.

Temperatures only work for adding and subtracting, but debt is a good analogy for multiplying and dividing - e.g. if you've got two £5 debts then you've got a £10 debt.

4. ### Vince_UlamStar commenter

Three questions:

1. Are you able to do what you are trying to teach?
2. If yes, then what is the obvious difference between your ability to do it now and your prior inability?
3. If no, then why are you trying to teach maths?

5. ### mon_sterNew commenter

I can teach students how to calculate with negative numbers but I am looking to help pupils develop complete fluency so as to minimise careless errors that can slow down their progress in later work, such as solving equations.

6. ### Vince_UlamStar commenter

Good evening, mon_ster.

Are you able to do what you are trying to teach?

7. ### mon_sterNew commenter

Yes Vince, ta for the help.

8. ### Vince_UlamStar commenter

Super, then you have obviously spent more time at it than your students. There is no magic trick to this. To become fluent at anything requires repetition.

The strategy for becoming proficient in the calculation of integers:

1. Practice.

You are welcome.

Last edited: Oct 9, 2015
9. ### mon_sterNew commenter

OK practice, that's a good start. When I asked about a strategy I was not looking for a silver bullet I was looking for advice on how frequently to revisit and to what intensity, 10 questions every lesson, a whole lesson every term etc.

10. ### Colleen_YoungOccasional commenter

Yes practice - keep revisiting. I find money works well. And number patterns - look at a whole multiplication table. Common sense - surely 2x3 need sto be different from 2x -3.
For some online resources have a look at some of these.

11. ### Vince_UlamStar commenter

Mon_ster.

You are right to want your students' skills to be automatic so here is your strategy anticipating your goal of working with equations: Carousel pen & paper practise of directed number, of fractions, of substitution and of operational order, focussing upon instrumentality not conceptual understanding.

• One session a week.
• First circuit of carousel consists of one full lesson opening each topic; 1/3 demonstration, 2/3 exercise.
• Crisp demonstration, accessible but precise mathematical language, no metaphor, no analogy.
• Whiteboard to be used only for free hand calculation and display of exercise.
• No long explanations at desks; students move on to next question if they have real difficulty while you take note; mark afterward and recover at opening of next session but do not attribute errors to students.
• Subsequent sessions include at least fifteen minutes of solid pen & paper exercise.
• After a minimum of three full circuits present exercises combining topics increasing in difficulty with time.

The above may not look to some people a pretty prospect but you want as little as possible between the students and the maths. You don't want them having to interpret whiteboard animations, work round new formats, think about a novel representation of number, become distracted by games or watch you struggling with a clunky menu or props. They need to be able to devote all of their cognitive capacity to acquiring the raw maths. Routine pen and paper practise is what works to secure these elementary skills and the best practise they can get is via pen & paper.

Last edited: Oct 9, 2015
12. ### mon_sterNew commenter

I can see this working for me, I appreciate the advice, thanks for taking the time to post.

13. ### Vince_UlamStar commenter

You are welcome.

14. ### pittanNew commenter

I use Starters as means to revise basic Maths skills. I normally use Ten Quick Questions for this. Proficiency does improve over time.

15. ### lou1990louNew commenter

I teach negative numbers as a whole lesson, set homework on it and feedback on their ability as a specific topic.
I then make sure that my starts for other topics use negatives, and these are often the question 3/extension questions.