Errors and Common

Misconceptions in the

Classroom KS3 & KS4

Dr Audrey Curnock

Director of Education Unlimited

June 29th 2016

1

Errors & Misconceptions

• Today’s talk is about trying to categorise errors –with the aim that giving supportive feedback we can encourage a deeper understanding of Mathematics

• and a more secure pathway in Learning Mathematics – towards Mastery!

June 29th 2016 2

Errors & Misconceptions

• When we look at learners’ work we try to see if the solution are efficient, methodical, clear, accurate, based on sound mathematics. …..And of course (mostly) correct!

(….it’s not the answer but the journey!)

• Learners’ solutions do not always bear this hallmark!

• As teachers we want to locate errors in learners’ work – Why? to understand what’s going wrong in their thinking/understanding. We all make small errors, often showing a lack of accuracy. This is not in itself a major problem, of course it needs to be addressed.

• But errors could contain or display misconceptions or indicate that the topic has not been grasped or understood at all.

June 29th 2016 3

Errors could be…– calculation, or wrong values have been taken – or miscopied

- mis-applied a piece of theory, wrong operation applied (eg KS2/3 dividing

when they should be multiplying)

- Muddled topics- wrong theory has been used – eg use Pythagoras but there’s

no right-angle.

- The strategy adopted leads nowhere. (A different strategy was needed) -

Circular arguments

- Reasoning has gone wrong (wrong deductions made during the question)

eg in GCSE A* questions when two-stage problems are used such as the

sine and cosine rules, and the logic has gone astray during these stages.

June 29th 2016 4

4

• WHAT ? : Adopt teaching strategies to address any substantive errors

• Use examples, question their thinking, address logical issues, address lack of knowledge or mis-learnt knowledge, or gaps in knowledge (it may take time to unravel where the thinking has gone wrong). Eg Paige – year 9 student. Any others?

• WHY? : It’s one of the principles of effective teaching “Expose and discuss common misconceptions.” (Swan, cf ILIM PD1).

• Research shows : (a)Teacher Orientation towards Student Performance has some influence on improving learning & also overall school improvement.(cfHoningh & Hooge & others, 2014).

• (b) Errors are essential when developing understanding (Ingram, Baldry, Pitt, ATM Magazine, May 2014) June 29th 2016 5

As teachers what can we do to address this?

– why should we? - and how will it help?

How will it help?

• How will it help ? New learning will not be embedded properly - In our TASK – will develop your thinking on this.

• If a learner has gaps in their knowledge this needs to be addressed.

• Mathematical learning develops as links in a chain.

• Topics are inter-connected. When we learn a new topics this needs to be build on a solid foundation, or it will not be absorbed /retained (or learnt!)

• Psychology of learners – be mindful of this – learners don’t want their errors exposed for their peers to see, they may become defensive – and learning stops.

* Use Errors as a discussion point (Ingram, Baldry & Pitt, ibidem)

June 29th 2016 6

How will it help?

-What sort of feedback helps?

• Feedback needs to be corrective in nature; tell students how they did in relation to specific levels of knowledge. Rubrics are a great way to do this.

• Aim to give feedback in a timely fashion and be specific.

• Encourage students to lead feedback sessions.

• These strategies are from “Classroom Instruction That Works” by Robert Marzano, Debra Pickering, and Jane Pollock. (2001)

• Let’s look at a couple of video clips

• https://www.youtube.com/watch?v=MzDuiqaGqAY & https://youtu.be/4hy-tCkLdy8

June 29th 2016 7

Feedback – Ego Vs Task

June 29th 2016 8

https://www.youtube.com/watch?v=MzDuiqaGqAY

Listening – individual feedback -Peer support

June 29th 2016 9• https://youtu.be/4hy-tCkLdy8

Policy.Profession@education.gsi.gov.uk 10

Accelerating students learning- what's the impact of various approaches? Effect Sizes here have been converted to months of school progress. Ref: http://educationendowmentfoundation.org.uk/toolkit/about-the-toolkit/ research funded by The Sutton Trust

Approach Average Impact in months

Behaviour interventions 4

Collaborative Learning 5

Digital technology 4

Feedback to students 8

Homework (secondary) 5

one-to-one tuition 5

Meta-cognition and self-regulation 8oral language interventions 5

Peer Tutoring 6

0

1

2

3

4

5

6

7

8

9

Average impact in months

Average Impactin months

Example 2 : Effect Sizes *

June 29th 2016

This is an extract from a workshop we did for the DfE – the report is in public domain

Examples of what examiners say about errors

• GCSE F

• Arithmetic often poor, algebraic manipulation below the standard required. Area and perimeter confused. Computations with fractions are difficult. Muddling acute and obtuse angles. Negative numbers pose a problem. Finding 42% of a value problematic.

• GCSE H

• Standard of arithmetic overall was poor, lack techniques of 4 basic rules. Presentation of answers was good, written communication clearly taught.

• Fractions, expanding brackets, confusion over time calculations, km-miles conversion all display problems. In the calculator paper – rounding prematurely. Tree diagrams –most could do – but weaker students couldn’t use them. Lack of confidence working with vectors. Use of random number tables. Standard form questions with no workings – and outside acceptable range.

June 29th 2016 11

Case Studies

The room is arranged into Key Stages

June 29th 2016 12

• TASK

• In groups of 3 or 4 look at the solutions you’ve been given for this range of problems

• Problems are from KS3, KS4 (IGCSE & GCSE)•

•

TASK CTD

June 29th 2016 13

Student Topic Identify the error – and

what type – is it a

serious error?

Feedback you would give

to improve the work and

progress in learning.

On the flip chart paper – draw 3 columns with these headings

For each topic add a Post-it Note saying what teaching

strategies would you adopt to overcome these difficulties

Conclusions

• Myriad of errors and we use a wide variety of teaching strategies to

help.

• Above all, Maths teachers are very much needed!!!

June 29th 2016 15

Honingh M & Hooge E, (2014) “The effect of school-leader support and participation in decision making on teacher collaboration in Dutch primary and secondary school”, EMAL journal, Vol 42, pp75-98.

Ingram J, Baldry F, Pitt A, (2014) “What role do errors have in the learning of Mathematics?” ATM Magazine, May 2014

http://educationendowmentfoundation.org.uk/toolkit/about-the-toolkit/

Marzano R., Pickering D., & Pollock J., (2001) “Classroom Instruction That Works,”

Video clips

https://youtu.be/4hy-tCkLdy8

PD1 & PD2 National Stem Centre e-documents, Standards Units – Improving Learning in Mathematics

June 29th

2016

16

https://www.youtube.com/watch?v=MzDuiqaGqAY

References

Notional Aword Holder for Mothemotics Tutor ol the Year 2013

Schools' Educational Consultancywww.education-un limited.co. u k

Su pporti ng th rivi n g Ed u cationo I

Cammunities

Key Stage 4 IGCSE Errors or Misconceptions(Use of a Calculator is available for all questions)

Example 1

The length of an Airbus A300 aeroplane is 54 m. The ratio of the length of this aeroplane to its wingspan

is 6 : 5. Work out the wingspan of the aeroplane.

Z.@Edexcel

Aiyo's Solution .,b

JJ_6

Example 2

Tickets for a show are priced at f 12.50 for an adult and f7.20 for a child ticket. Find

the total cost, in f, for 5 adults and 4 child tickets.

OEdexcel

Harry's Solution

I Z.9o + 7.Zo : 1.V" t s - {+".s

(-^ tJ = 5+'- 2.

-{t+

f

Is4 qjjl

@ Education Unlimited Education Unlimited is a Division

Ciili;Iuil]g Pfoiilssicriarl

iJilvelopilcii('dl

'": r'

National C.n,r" *'jr :.a4, -'i:.:: r: Iiri

Page tr

i;-rFi ri:rg ii".i.1a J:Drsrii ri ili:i-"4'fi,

t{a?!aliiia:: tiI::i:$ l

t-t)\tx)\\1.{r'llr.}i,il r{:Al

of Aspiring Heads Consulting LTD.

National Awdrd Holder for Mdthemotics Tutor of the Year 2013

Schools' Educational ConsultancY

www.education-unlimited.co.ukSu pporti n g th riv i n g Ed u catia na I

Communities

Example 3

ln the diagram AeB is parallel to CRD and PQRS is a straight line. Find the value of x.

@Edexcel

Nikhat's solution

Ka 2r-*+o Z sLrr& ^il L"-3a+{erR:(&o (YLr- o\.\- c^- l.j"--

Wri, G"[I

DevelopmentSta0dard

National Centrefnr Eraellerig li ille

i: ; :i,'!

(3x + 10)"

2r' 4o*1o

@ Education Unlimited

L()\D()\H.\ IltIlt{l'l( ]At-

5o

-ra*oi.r,

,l"ri.i "Ji,'u'oiutiion

6iEiFiring Heads Consulting LTD.

Continuinq Professiorlal

Page 2

AsGiri*g Heads Ccnsulting

LTD,

Registered offiee :

Nationol Award Hotder lor Mathematics Tutor of the Year 2073

Schools' Educational ConsultancY

www.education-unlimited'co.uk

Example 4

ABCD is a rectangle.

Leaving allyour construction lines in,

Su ppo rti n g th riv i n g E d ucati ona I

Communities

Lb)

the locus of all points inside the rectangle which are

a)

blEguidistant from D and C,

Equidistant from the lines AD and DC'

The region R consists of all the points inside the rectangle which are closer to

AD than DC.

cl Show by shadingthe region R' Labelthe regicn R'

OEdexcel

Rosalind's solution d,^ S* J,l.y*ur,' olto,<J

@ Education Unlimited Education Unlimited is a Division of Aspiring Heads Consuhing LTD'

Continuing Pro{essional

Development

o-*rr*t*'u*

is

C than D and closer to

Page 3

Aspiring F{eads Constriting

LTD,

Eeaistered cff!ce :t:i

'liff

i-::rri:i Lt)ID()\:t ;i' t't.{t Htl}L'rf lCitl-i-:- rl