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Hiding in the bookshelves in the University of Otago Library, I wept as I read the sentence, “There are many good ways to raise children.” As a mother of a baby with severe disabilities the burden to get it right weighed down on me. This statement told me to put down the burden. I could do things differently from other mothers, and none of us needed to be wrong.

The same is true of teaching maths and stats – “There are many good ways to teach mathematics and statistics.” (Which is not to say that there are not also many bad ways to both parent and teach mathematics – but I like to be positive.)

My previous post about the messages about maths, sent by maths and stats videos, led to some interesting comments – thanks especially to Michael Pye who “couldn’t get the chart out of [his] head”. (Nothing warms a blogger’s heart more!). He was too generous to call my description of the “procedural approach” a “straw-person”, but might have some justification to do so.

His comments (you can see the originals here) have been incorporated in this table, with some of my own ideas. In some cases the “explicit active approach” is a mixture of the two extremes. The table was created to outline the message I felt the videos often give, and the message that is being encouraged in much of the maths education community. In this post we expand it to look at good ways to teach maths.

Procedural approach | Explicit but active approach | Social constructivist approach | |

Main ideas | Maths is about choosing and using procedures correctly | Maths is about understanding ideas and recognising patterns | Maths is about exploring ideas and finding patterns |

Strengths | Orderly, structured, safe, cover the material, calm | Orderly, structured, safe, cover the material, calm and satisfying | Exciting, fun, annoying |

Skills valued | Computation, memorisation, speed, accuracy | Computation, memorisation, (not speed), accuracy + the ability to evaluate and analyse |
Creativity, collaboration, communication, critical thinking |

Teaching methods | Demonstration, notes, practice | Demonstration, notes, practice, guided discussion and exploration via modelling. | Open-ended tasks, discussion, exploration |

Grouping | Students work alone or in ability grouping | Students discuss as a whole class or in mixed-ability groups | |

Role of teacher | Fount of wisdom, guide, enthusiast, coach. | Fount of wisdom, guide, enthusiast, coach. | Another learner, source of help, sometimes annoyingly oblique |

Attitude to mistakes | Mistakes are a sign of failure | Mistakes happen when we learn. (high percentage of success) | Mistakes happen when we learn. |

Challenges | Boredom, regimentation, may not develop resilience. | Boredom, regimentation, could be taught purely to the test | Can be difficult to tell if learning is taking place, difficult if the teacher is not confident |

Who (of the learners) succeeds? | People like our current maths teachers | Not sure – hopefully everyone! | |

Use of worksheets and textbooks | Important – guide the learning | Develops mastery and provide assessment for learning. Limits gaps in understanding. | Occasional use to supplement activities |

Role of videos | Can be central | Reinforce ideas and provide support out of class. | Support materials |

We agree that speed is not important, so why are there still timed tests and “mad minutes” .

The previous post was about the messages sent by videos, and the table was used to fit the videos into a context. If we now examine the augmented table, we can address what we think good mathematics teaching looks like.

The biggest question when discussing what works in education is “for whom does it work?” Just about any method of teaching will be successful for some people, depending on how you measure success. Teachers have the challenge of meeting the needs of around thirty students who are all individuals, with individual needs.

I have recently been considering the scale from introvert – those who draw energy from working alone, and extraversion – those who draw energy from other people. Contrary to our desire to make everything binary, current thinking suggests that there is a continuum from totally introverted to totally extraverted. I was greatly relieved to hear that, as I have never been able to find my place at either end. I am happy to present to people, and will “work a room” if need be, thus appearing extraverted, but need to recover afterwards with time alone – thus introverted. Apparently I can now think of myself as an ambivert.

The procedural approach to teaching and learning mathematics is probably more appealing to those more at the introverted end of the spectrum, who would rather have fingernails extracted than work in a group. (And I suspect this would include a majority of incumbent maths teachers, though I am not sure about primary teachers.) I suspect that children who are more extroverted will gain from group work and community. If we choose either one of these modes of teaching exclusively we are disadvantaging one or other group.

In New Zealand we are finding that children from cultures where a more social approach is used for learning do better when part of learning communities that value their cultural background and group endeavour. In Japan it is expected that all children will master the material, and children are not ability-grouped into lowered expectations. Dominant white western culture is more competitive. One way for schools to encourage large numbers of phone calls from unhappy white middle-class parents is to remove “streaming”, “setting”, or “ability grouping.”

I recently took part in a Twitter discussion with maths educators, one of whom believed that most maths classes should be undertaken in silence. One of the justifications was that exams will be taken in silence and individually. This may have worked for him, but for some students the pressure not to say anything is stifling. It also removes a great source of learning, their peers. Students who are embarrassed to ask a teacher for help can often get help from others. In fact some teachers require students to ask others before approaching the teacher.

As is often the case, the answer lies in moderation and variety. I would not advocate destroying all worksheets and textbooks, nor mandate frequent silent individual work. Here are some of suggestions for effective teaching of mathematics.

- Having variety in your approaches, as well as security
- Aiming for understanding and success
- Trying new ideas and having fun
- Embracing your own positive mathematical identity (and getting help if your mathematical identity is not positive)
- Allowing children to work at different speeds without embarrassment
- Having silence sometimes, and noise sometimes
- Being competent or getting help – a good teaching method done poorly is not a good teaching method

Here are links to other posts related to this:

The Golden Rule doesn’t apply to teaching

Educating the heart with maths and statistics

The nature of mathematics and statistics and what it means to learn and teach them

And thank you again to those who took the time to comment on the previous post. I’m always interested in all viewpoints.

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## 2 Comments

Cognitive load theory supports quick timed test for recall work. The trick is the success rate to make it not stressfull. Tests aren’t inherently stressfull but doing badly or fearing doing so certainly is. Unfortunately designing these is hard and far to many of us have to make them from scratch. Software really can offer something here or alternatively something like ninja numeracy which offers each student enough question types to experience success.

I used to be dubious of this kind of thing as it always needed to much scaffolding but when I realised it was just mental excercise to get students recall warmed up ready for later work I was really convinced. The timing is also easily adjusted. I often use this with my SEN groups with a generous time allowance, some sneaky peer support and a few examples scribbled on the whiteboard. Great starter or finisher.

What you say may be true, but when I talk to people about the traumatic experiences they have had with maths, the vast majority come back to timed tests. Maybe it is possible to do this without harming people, but I believe there are better, less risky approaches.