Singapore Maths and the Struggling Learner

Singapore Maths and the Struggling Learner
it will detail what dyscalculia is and how the Singapore method can support these learners . The talk will draw on research from Bruner, Dienes and Vygotsky, amongst others and will give practical strategies for teaching learners who are struggling with maths. Judy Hornigold

Contents Why learners struggle with maths
Different forms of dyscalculia Singapore maths Other strategies to help Judy Hornigold

Why can maths be so hard? Cumulative Perceptions Performance focus Timed Mental maths Too much content Too abstract Judy Hornigold

What are the different forms of dyscalculia?
Acalculia General difficulties with maths Dyscalculia Pseudo-dyscalculia Paul Beau General difficulties = general difficulties with all learning – a generally lower profile of performance across all subject areas Dyscalculia = normal intellectual capacity, spikey profile, automatisation difficulty, difficulty with planning, problems with visual perception, problems with memory & language, sequencing Pseudo-dyscalculia = emotional blockage (girls form the largest part of this group) dealing with this one would be the same as dealing with dyscalculia, yes its an emotional response but due to increase chemicals in the body caused by extreme anxiety, e.g. adrenalin, (fight or flight response) etc it does mean that individuals who experience this sort of reaction are potentially not in a suitable emotional state to learn effectively. What do they think about this one – does it mean there is potentially a case for teaching boys & girls separately (only a discussion point but causes some interesting debate!) Judy Hornigold

What is dyscalculia? How is it different from being bad at maths?
Dyscalculia is a specific learning disability (SpLD) that affects a person’s ability to acquire arithmetical skills. People with dyscalculia find it hard to understand basic number concepts and/or number relationships, recognise symbols, and comprehend quantitative and spatial information. Many people liken the effects of dyscalculia with numbers to that of dyslexia with words, and while there are many characteristics that overlap, there is no proven link between the two. These difficulties can have an adverse effect on many day-to-day activities such as dealing with finances, following directions, managing a diary and keeping track of time. However, it is important to remember that many people can struggle with maths and numbers, but this does not mean that they have dyscalculia. This is an important point to explore here- what is the difference between dyscalculia and having difficulty with maths generally It is estimated that between 4% and 6% of the population suffer with dyscalculia. The National Numeracy Strategy (DfES, 2001) offers the following definition: ’Dyscalculia is a condition that affects the ability to acquire arithmetical skills. Dyscalculic learners may have difficulty understanding simple number concepts, lack an intuitive grasp of numbers, and have problems learning number facts and procedures. Even if they produce a correct answer or use a correct method, they may do so mechanically and without confidence.’ (DfES, 2001, p2). Judy Hornigold

Definitions of Dyscalculia
Dyscalculia is a condition that affects the ability to acquire arithmetical skills. (DfES 2001) A congenital condition: its effects on the learning of numerical skills can be very profound. (Butterworth) Dysfunction in the reception, comprehension or production of quantitative and spatial information (Sharma). Discuss each of these definitions in turn- what are the implications of each? Butterworth takes a more neurological approach- if it is a brain difference then what implication might this have in terms of intervention? Judy Hornigold

The National Numeracy Strategy DfES (2001)
Dyscalculia is a condition that affects the ability to acquire arithmetical skills. Dyscalculic learners may have difficulty understanding simple number concepts, lack an intuitive grasp of numbers, and have problems learning number facts and procedures. Even if they produce a correct answer or use a correct method, they may do so mechanically and without confidence Key Features “acquire” emphasises acquisition rather than carrying out arithmetic procedures. “difficulty understanding simple number concepts, lack an intuitive grasp of numbers” placing understanding at the core of dyscalculia “A lack of a true comprehension or understanding of maths will be a key characteristic of dyscalculic people” Chinn S. (2006) “Learning number facts and procedures” : more dyslexia related? Judy Hornigold

Mathematics Disorder:
DSM-IV (2000) Mathematics Disorder: "as measured by a standardised test that is given individually, the person's mathematical ability is substantially less than would be expected from the person’s age, intelligence and education. This deficiency materially impedes academic achievement or daily living" Judy Hornigold

struggles to understand mathematical concepts – speed, time etc.
difficulties with telling the time left/right confusion cannot accurately recall number facts map reading difficulties problems handling money – working out change etc. constantly re-learning and recapping skills issues with place value organisational issues difficulty navigating back and forth along a number line or sequence struggles to understand chronology 123 Go through each of these areas and flag up the TES resources as a great place to access resources for dyscalculia problems transferring information: e.g. 5+4=9 therefore 4+5=9 can lose place easily lack confidence in their answers finds counting in twos, threes etc. problematic Judy Hornigold

Indicators of Dyscalculia
An inability to subitise even very small quantities Poor number sense An inability to estimate whether a numerical answer is reasonable Immature strategies- for example counting all instead of counting on Inability to notice patterns Inability to generalise Subitising comes from Latin= ‘Sudden’ It is the ability to give the amount of objects in a set without counting Most people can subitise up to five or six objects Dyscalculic people do not have this ability- they may not even be able to subitise a set of two items Is this innate? Are we born with the ability to assess quantity? Yes, being able to identify quantity is a survival instinct and is innate in us all- but people with dyscalculia may not have this ability Number sense The ability to determine the number of objects in a small collection, to count, and to perform simple addition and subtraction, without direct instruction. Spoken language and number sense are survival skills but abstract maths is not. Go through each of these indicators and discuss with the group; Do they recognise these difficulties in any of their learners? Why and how would these difficulties impact on their ability to learn maths Main ones to highlight are- inability to subitise, persisitance in using immature strategies, inability to notice patterns, and inability to estimate. Judy Hornigold

Indicators of Dyscalculia (cont’d)
Slow processing speed Difficulty sequencing Difficulty with language Poor memory for facts and procedures Difficulties in word problems and multi step calculations Judy Hornigold

How do you identify and assess for dyscalculia?
Dyscalculia Screener- Butterworth Dynamo Profiler DyscalculiUM- FE/HE screener Questionnaire/Checklist Observation Judy Hornigold

DYSCALCULIA SCREENER(Nfer/Nelson 2003)
Developed by Brian Butterworth Based on research that showed dyscalculic pupils performed worse on certain numerical processing skills Tests include: Dot counting Number comparison Timed arithmetic GL assessment- show demo Judy Hornigold

Checklists BDA Dyscalculia Checklist
Hand out and go through the BDA dyscalculia checklist Judy Hornigold

Observation In class Observe how they attempt a question Look for signs of stress Encourage the learner to verbalise how they are attempting the maths One of the most informative ways to assess for dyscalculia is to observe the learner in a maths class, ask them to verbalise to you how they are doing the maths and what strategies they are using Judy Hornigold

A learned response Anxiety rises when faced with mathematics and can be so severe that even the sight of a maths problem can lead to paralysing anxiety Fight or flight reaction Explain- adrenalin, pavlovian response Judy Hornigold

Physical Symptoms of Maths Anxiety
Kay Haralson October 11, 2002 NCTM Physical Symptoms of Maths Anxiety queasy stomach, butterflies clammy hands and feet increased or irregular heartbeat muscle tension, clenched fists tight shoulders Judy Hornigold Austin Peay State University

Physical Symptoms of Maths Anxiety
Kay Haralson October 11, 2002 NCTM Physical Symptoms of Maths Anxiety feeling faint, shortness of breath headache shakiness dry mouth cold sweat, excessive perspiration Judy Hornigold Austin Peay State University

Psychological Symptoms of Maths Anxiety
Kay Haralson October 11, 2002 NCTM Psychological Symptoms of Maths Anxiety negative self-talk panic or fear worry and apprehension desire to flee the situation or avoid it altogether a feeling of helplessness or inability to cope Judy Hornigold Austin Peay State University

Maths Anxiety can be related to
Life Experiences attitudes of parents, teachers or other people in the learning environment some specific incident in a student’s math history which was frightening or embarrassing poor self-concept caused by past history of failure Learner’s attitudes Lack of belief in ability Lack of faith in intuition Negative beliefs Giving up before really trying ‘Dropped stitch’ thinking Depression Fear of failure Judy Hornigold

Maths Anxiety can be related to
Teaching Techniques Anxiety can be caused by teaching techniques which emphasize: timed activities the right answer speed in getting the answer competition among students working in isolation memorization rather than understanding Judy Hornigold

Maths lessons “Evidence suggests that maths anxiety results more from the way the subject is presented than from the subject itself.” J. Greenwood (1984) Few maths classes are structured in such a way as to relieve anxiety. There will always be time limits, right answers, and competition. Reducing maths anxiety will not make students ‘smarter’ in maths. However, it could allow a students to reach their full potential Judy Hornigold

Ways teachers can help Foster the idea that mistakes are good
Spend time developing number sense at an early age Encourage a relaxed participatory atmosphere Use concrete manipulatives in a way that develops understanding We will spend the rest of the talk looking at each of these in turn Judy Hornigold

Mistakes ‘Every time a student makes a mistake in maths they grow a new synapse’ Carol Dweck No mistake – no growth Judy Hornigold

Mistakes matter Mistakes are vital in maths
New data has shown that the brain grows when a mistake is made No growth when the answer is correct If the answer is correct then there is no growth- the child does not even have to know why they got it wrong and they don’t have to go back and get it right- it is the struggle that counts Most children feel really bad if they make a mistake , but they should feel that they are learning and that it will help them This knowledge could go a long way to helping overcome maths anxiety Judy Hornigold

Mistakes are good Move from a performance culture to one where tasks are open and promote growth Give challenging work Value persistence Children to need to think that they are there to enjoy, learn and grow For children with maths anxiety Screw up some paper into a ball and throw it at the board to represent how they feel when they cant do maths- then retrieve their paper and unscrew it- flatten it out and look at all the creases- these are the new synpases that have grown in the brain when the mistake was made or when you were struggling We want children to feel comfortable making mistakes Mistakes are when children learn the most Students need challenging complex work Environment where mistakes and persistence are valued Judy Hornigold

Implications Each child has enormous potential to grow their brain, no matter where their starting point is Teachers have the power to take children to high levels Brain difference at birth has minimal impact on future learning- it is nurture rather than nature that matters. Judy Hornigold

Mindset How important are the ideas that students hold about their own ability? Judy Hornigold

Carol Dweck: Mindset Fixed mindset: Maths ability is a gift
Growth mindset: Maths ability grows with experience , persistence, learn from mistakes, determination to keep going, encouraged by other’s success Mindset: The New Psychology of Success (2007) Carol Dweck has trained premier league footballers If your mum says ‘I wasn’t any good at maths then the child’s marks will go down’ Fixed mindset comes from fixed praise If a parent says you are clever, then the child will hear that , but this can backfire later if they struggle with something because they think that this means they are not clever Better to say you have tried really hard. You have done really well Experiment- half of a class were given one extra sentence in their feedback from teachers and the other half did not have that sentence. The chidlren with the extra sentence did better when tested one year later The sentence was I am giving you this feedback because I believe in you Judy Hornigold

Mindset and Maths Lowest achieving children are those who use a memorization technique for maths Highest achievers are those who look for connections and have big ideas Lowest achieving are actually doing the hardest maths Judy Hornigold

Does it matter how quickly we can answer?
Research evidence shows that maths should never be associated with speed Timed tests cause the early onset of maths anxiety for about 1/3 of the children in the class Being fast at maths is not the same as being good at maths Most mathematicians think slowly and deeply We have a mile wide and inch deep curriculum- it should be narrower and much deeper ( reference the ks2 mental maths) Professors of maths often don’t know their times tables- Jo Bolaer doesn’t Judy Hornigold

How to help Singapore Maths Problems and Solutions Using dot patterns
Sharma’s levels Key facts and derived facts Judy Hornigold

Singapore Rises to the Top
TIMSS first conducted in 1995 From 1982 – 1995 Singapore rose to the top In 1999, first in math, but second in science Trends in International Math and Science Study (TIMSS) Judy Hornigold

make note of 94% in Intermediate or higher.
TIMMS Benchmark 2011 Grade 4 is the same age as UK Year 5 Since 1995 Singapore has been at the top of mathematics education make note of 94% in Intermediate or higher.

TIMMS Benchmark 2011 Grade 8 is the same age as UK Year 9 Singapore bucks the trend and maintains its high results in Secondary school. make note of Singapore bucking the trend and holding results in secondary schools

TIMMS Benchmark 2011 Grade 8 is the same age as UK Year 9 Singapore used to be part of Malaysia and previous to changing how they teach in the 1980s their results were identical to Malaysia Make note that Singapore used to be part of Malaysia and used to mirror their results until 1980s when they changed education system.

Curriculum is articulated in a pentagonal model
Math curriculum in the US has done well with developing skills and concepts and to some extent processes, but lacked in metacognition and attitudes. Heurisitcs experience-based techniques for problem solving, learning, and discovery that finds a solution which is not guaranteed to be optimal, but good enough for a given set of goals. Where the exhaustive search is impractical, heuristic methods are used to speed up the process of finding a satisfactory solution via mental shortcuts to ease the cognitive load of making a decision. Examples of this method include using a rule of thumb, an educated guess, an intuitive judgment, stereotyping, or common sense. Judy Hornigold

Big Ideas in Singapore Math
Number sense Making connections and finding patterns Communication Visualization Concrete Pictorial Abstract Variation Number sense implies having a deep understanding of mathematical concepts, making sense of various mathematical ideas, as well as developing mathematical connections and applications Students learn to make linkages among mathematical ideas, between mathematics and other subjects, and between mathematics and everyday life. Students learn to use mathematical language to express mathematical ideas and arguments precisely, concisely, and logically. Back in the late 1980’s NCTM published articles about best practices in mathematics instruction which included C-P-A Students practice problems presented in many different ways, rather than many problems in the same way. Judy Hornigold

Number Bonds/ Splitting Numbers
Part-part-whole representation Introduced in K Used in 1 as a tool for addition and subtraction In the upper grades, used to decompose numbers when demonstrating mental math strategies “What’s your number?” Judy Hornigold

Singapore Maths Encourages algebraic thinking even among early learners Reduces complexity Promotes deeper understanding Enhances problem solving Helps all students Supports struggling learners Extends gifted Judy Hornigold

CPA Model Jerome Bruner
Enactive Symbolic Iconic Links back to Sharma talk from yesterday Judy Hornigold

Bar Modelling Judy Hornigold

Example There are 9 white flowers. There are 3 times as many red flowers as white flowers. How many red flowers are there? Two quantities are compared. One is a multiple of the other. We know the smaller quantity. To find the bigger quantity we multiply 9 x 3. Judy Hornigold

Problem Solving See Plan Do Look Back
“Bar modeling allows students to engage in algebraic thinking even before they are ready to handle formal algebra.” -Bar Modeling A Problem Solving Tool, Yeap Ban Har, PhD See: Visualize the problem – retell without numbers Determine what is known and what needs to be answered Plan: Determine the appropriate operation or operations needed Look for connections to similar problems Do: Carry out the plan Determine if revisions or modifications are necessary Check for reasonable-ness Look Back: Was the question answered? State the answer in a sentence Reflect – Could I have solved the problem more efficiently? Judy Hornigold

Heuristics

Singapore Maths A Typical Lesson Mental math activities
Hands-on lessons with manipulatives Partner conversations Guided and independent practice Differentiation 3 different styles of learning: kinesthetic, visual, and auditory Depth of knowledge within the same topic Judy Hornigold

What makes Singapore Math such a strong curriculum?
• Emphasizes the development of strong number sense, excellent mental‐math skills, and a deep understanding of place value. • A progression from concrete experience using manipulatives, to a pictorial stage, and finally to the abstract level or algorithm. • Gives students a solid understanding of basic mathematical concepts and relationships before they start working at the abstract level. Judy Hornigold

Includes a strong emphasis on model drawing, a visual approach to solving word problems that helps students organize information and solve problems in a step‐by‐step manner. Because pupils work with concrete apparatus, they are always solving problems rather than just learning algorithms – they become good problem solvers Judy Hornigold

Principles ‘Five Wise Guys’- Dr Ban Har Yeap
Piaget- Teach less – learn more Bruner- CPA Dienes- Informal before formal Skemp- Relational Understanding Vygotsky- Collaboration Judy Hornigold

Using Dot Patterns Dot patterns bridge the gap between concrete and abstract work Will help develop a sense of number Will help develop the concept of conservation of number Can be linked to familiar patterns eg dice or dominoes Using dot patterns is a great way to develop an understanding of number and to help dysclaulic learners to compare quantities. It will also develop there ability to subitise greater quantities Dot patterns sheet Judy Hornigold

Key facts and Derived facts
Consider the UK money system 1p, 2p, 5p and 10p Why have we chosen these amounts? We can make any number from the different amounts- so 1,2 ,5 and 10 are considered key facts. We can support learners with dyscalculia by focussing on key facts and modelling how to derive new facts from things they already know. This links well to the principle of using what you know to move through the six stages to find out what you don’t know We will look at how we can reduce the memory load for children when learning the tables– using coins can also be a great support when teaching tables Judy Hornigold

Key Facts Key Facts are the ‘easy know’ facts
The same for every times table 2xn= 5xn= 10xn= Judy Hornigold

Key facts Multiplication by repeated addition The ‘Key Facts’ are: 1x 2x 5x 10x 3 x 8 = x8 and 2x8 6 x 8 = x8 and 5x8 7 x 8 = x8 and 5x8 12 x 8 = x8 and 10x8 Cuisennaire rods can be used here to model the multiplication arrays- and then you can go through the stages in turn Number strips and number squares are also useful Judy Hornigold

Derived facts If we know that 5+5=10 then what else can we derive from this? 50+50= =11 5+4=9 Judy Hornigold

Derived Facts 72 students aged 7 – 13 years… addition Above average:
9% counted on • 30% known facts • 61% derived facts Below average: 72% counted on • 22% counted all • 6% known facts • 0% derived facts Gray and Tall University of Warwick Judy Hornigold

Making Links We need to show children how to make links . Many areas of maths are closely linked and it will help undertsanidng if these links are explicitly demonstrated As well as writing sums in the form 4+5=9 we need to write it the other way around too 9=4+5 as some children are completely thrown when the answer comes first Judy Hornigold

Sharma Four major principles for teaching
Use of Appropriate Concrete Models 2. Levels of Knowing Mathematical Ideas 3. The Three Components of a Mathematical Idea 4 The Questioning Technique Professor Sharma Berkshire Mathematics We will go through each if these in turn Professor Sharma’s website is Judy Hornigold

1.Use of Appropriate Concrete Models
For early mathematical concepts, it is important that a child experiences mathematics through appropriate and efficient learning models. Cuisenaire rods, base 10 materials and the Invicta Balance provide appropriate models for these concepts. It is important not to take away the concrete materials too early Fingers are good as a tool to help.- though we need to move away from reliance on them When you are doing basic maths the greatest activity in the brain is in the left parietal lobe and the region of motor cortex that controls the fingers- so our brains are wired to use our fingers when performing calculations Judy Hornigold

2. Levels of Knowing Mathematical Ideas
Intuitive Concrete Pictorial Abstract Applications Communication We need to work through each of these levels every time or the child will be stuck at the concrete or abstract level- with out the understanding to enable reasoning and learning Intuitive Every new fact is introduced as an extension of something the child already knows – students with dyscalculia are likely to have such connections dominated by their knowledge obtained through real life and literacy. Thye are likely to need guidance in making connections from this to what they are being taught Concrete Each new fact is presented through a concrete model and this will help the student to visualise and understand the underlying mathematical concepts Pictorial The model of the new fact may be sketched or illustrated. The student can then use this image to gain the answer. It is important to link language to the picture here as students may ignore accompanying text, or not read it thoroughly. Before abstract recording is asked for, a lot of oral and mental arithmetic activity is necessary. Abstract The new fact is recorded in symbolic form. e.g = 7 or by using diagrams or charts. It is important that the other stages are fully explored before embarking on the abstract stage- as many dyscalculics struggle with the abstract nature of maths Applications Intramathematical This is the ability to apply knowledge from one area of maths to another- eg to say that 0.5=1/2 Interdisciplinary This is the ability to apply maths to other subjects in the curriculum Extracurricula This is the ability to apply the maths to real life situations, eg to be able to double the quantities in a recipe The child is able to form a number story using the fact – this is a vital stage- developing mathematical language- working from the maths to the word problem and not the other way round which is what they are normally presented with. Communications The student is able to explain the strategy. e.g. Since = 6, I know is one more, so = 7 Generalising, reasoning and extending. Judy Hornigold

3. The Three Components of a Mathematical Idea
When presenting a mathematical idea we must make sure that equal weight is given to each of these three components Linguistic – is the language used in understanding,conceptualizing and communicating mathematical information. The linguistic component is the language (vocabulary, syntax, and translation from native language to mathematical language, and vice versa) used in understanding, conceptualising and communicating mathematical information. For example, to understand the concept of lowest common multiple, one needs to know the meaning of the individual words in the expression and their relationship to each other. In that sense mathematics is a language. 2) Conceptual The conceptual component is the mathematical idea itself. Modelling the idea (the concept) with concrete materials and manipulating these materials develops conceptual understanding. The child will then have a mental image of the concept to refer to. For example, conceptually what do the expressions 9 x 3 or ½ x 1/3, or (a)(b) mean? Each one of these expressions creates a physical or conceptual model in our minds. 3) Procedural – is the algorithm or the method emanating from the use of the concept, for example, the division algorithm or the procedure of adding fractions. Acquisition of the conceptual and linguistic components of mathematics will help the child to think mathematically, otherwise he/she will remain at a very procedural level. (Inchworm level) Mathematics will be a collection of tricks and remembering these tricks will become the learning of mathematics. Children will see mathematics as merely a collection of procedures if this aspect is emphasised without the use of concrete models and language to clarify the concept. Children very often forget the procedural aspect, but once the conceptual and language model is developed, it is easier to remember the procedure. Therefore, conceptual and linguistic understanding lies at the heart of learning mathematics. Judy Hornigold

The importance of mathematical language
In order to fully understand and grasp mathematical concepts it is vital that children are able to describe what they are doing using mathematical language before they attempt to put it into symbols. The more that we can relate the maths that they are doing to the real world the better. This makes it relevant and helps them to make sense of the world Mathematics is what we use to understand the world around us, the world is based on mathematical patterns and relationships. Our brains are hard wired to recognise pattern and this can be seen all around us in nature. Judy Hornigold

4. The Questioning Technique
For the development of concepts, the teaching process must engage the child by asking key questions. Appropriate questioning is important for the introduction of a concept, for reinforcing it and for helping the child to memorise facts. What kinds of questions are best? We want the child to discover the maths for themselves as they will learn more this way, but we need to ask questions that will draw out the answer that we want- Example What is the problem? · What do you know about this problem – concept, fact, formula, theorem? · Can you restate the problem in terms of what you know? · Have you solved another problem like this? · What is a possible solution? · How do you prove that this is a solution? · What is another possible solution? · What solution has the best chance of succeeding? Socratic questioning- The overall purpose of Socratic questioning, is to challenge accuracy and completeness of thinking in a way that acts to move people towards their ultimate goal. Conceptual clarification questions Get them to think more about what exactly they are asking or thinking about. Prove the concepts behind their argument. Use basic 'tell me more' questions that get them to go deeper. Why are you saying that? What exactly does this mean? How does this relate to what we have been talking about? What do we already know about this? Can you give me an example? Probing assumptions Probing their assumptions makes them think about the presuppositions and unquestioned beliefs on which they are founding their argument. This is shaking the bedrock and should get them really going! What else could we assume? Please explain why/how ... ? How can you verify or disprove that assumption? What would happen if ... ? Do you agree or disagree with ... ? Probing rationale, reasons and evidence When they give a rationale for their arguments, dig into that reasoning rather than assuming it is a given. People often use un-thought-through or weakly-understood supports for their arguments. Why is that happening? How do you know this? Show me ... ? Can you give me an example of that? Why? (keep asking it -- you'll never get past a few times) What evidence is there to support what you are saying? Questioning viewpoints and perspectives Most arguments are given from a particular position. So attack the position. Show that there are other, equally valid, viewpoints. What alternative ways of looking at this are there? Why is it better than ...? How are ... and ... similar? How could you look another way at this? Probe implications and consequences The argument that they give may have logical implications that can be forecast. Do these make sense? Are they desirable? How could ... be used to ... ? What are the implications of ... ? How does ... fit with what we learned before? What is the best ... ? Why? Questions about the question And you can also get reflexive about the whole thing, turning the question in on itself. Use their attack against themselves. Bounce the ball back into their court, etc. What was the point of asking that question? Why do you think I asked this question? Am I making sense? Why not? What else might I ask? What does that mean? Judy Hornigold

Summary CPA Approach Teach less and they will learn more Visualisation
Communication Number sense Problem Solving focus Judy Hornigold

Any Questions? Judy Hornigold

Singapore Maths and the Struggling Learner

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