A Balanced Mathematical Diet
Saturday 10 August 2013
As a teacher, if I can skilfully implement a wide range of approaches and methods, I’m most likely to satisfy the widest range of students’ needs
This year I have the mixed ability 14/15yr old group. Over two years we need to work together to achieve the goals students set themselves for their General Certificate of Secondary Education exams aged 16. For some, this may be the last years of formal Mathematics education as they move into apprenticeships, colleges or elsewhere. At this age, students often have a stronger idea of what they want from their mathematics education and I find myself needing to reflect in more depth about their individual needs and how best to meet them. The following is a reflection on the range of resources I use to help meet students’ different needs underpinned by the hypothesise (distilled from observation of great performers in different fields: sports, managers, leaders, performers i.e. the best performers can do everything) that if I can skilfully implement a wide range of approaches and methods I am more likely to satisfy the widest range of students’ needs.
The Scientific Method
Exploration, investigation and games are a great way to introduce a topic, exposing students to the key concepts, rather than the technical “how to” skill set, and requiring them to think it through for themselves: Experiment, Collect Data, Analyse the Data (looking for patterns), develop Hypothesise/Conjectures, Test those conjectures . . . . and maybe, in some cases, “Prove”. This ties in with the 1982 Cockroft report general finding of a greater need for a “sense of measure”. Industry and older generations are shocked when after five years of compulsory secondary education young people don’t realise that an answer of 121 x 24 = 290.4 must be wrong because it isn’t reasonable. Students need to have a range of estimation strategies to check the results of any algorithms used. They need to be trained to think about what they are doing beyond the mechanical application of “rules”.
Investigation, Exploration and Instant Feedback
This Straight Line Graphs resource first allows some instant feedback for students on whether they have understood how to define vertical or horizontal lines. Some explanation is likely to be required of how, and why, we define horizontal and vertical lines. This could be from the teacher, or using an “equations of horizontal and vertical lines” search on Youtube applying a “Short (~4mins)” filter or “view count”.
In a digital age, effective selection of search terms and filters is a great skill to be learning in any class.
It moves on to an investigation where you wouldn’t need to give any prior information to students as to how we define lines. By careful observation (experimentation), recording of results and looking for patterns (analysis) students can work out for themselves the effect of the gradient coefficient and y-intercept constants for themselves. In practice, not all students are likely to work these out, which offers a good example of “crowd sourcing” or the “hive mind” approach to solving problems: “no one individual of pair of people in this class may be able to solve the problem alone, but I guarantee that between all of you, you will have worked out for yourselves how an equation defines a line by the end of the lesson!”. Document Cameras are a great means of quickly sharing students' work with the whole class (alternatively you can take photos on your smartphone with your dropbox or icloud account open on the PC ready to project and share). The option to move onto the “bowling games” is a great motivator for students to find the equation of a line, encouraging engagement and effort. It provides a “need” for using equations to define lines and instant feedback and practice in a very enjoyable and engaging context. This resource has helped to impart understanding of the key concept that equations define a relationship for getting from the x-coordinate to the y-coordinate -students have to use thecoordinates of the bowling pins. Yet students can still forget this concept, despite having understood it and applied it with success during the activity. Moreover, when they see the same problem in a different context: a test paper with four lines drawn on a grid asking students to write the equation of each line, or the equation of a line written down with a grid for the student to draw the line, they can simply forget about equations and how they define the relationship between coordinates. What solutions present themselves in this case? We could revisit the bowling game to recap the key points, ask students as part of the homework following the bowling game to design a quick revision powerpoint, slideshare, video, song etc. and share these with the whole class – students can then select the video that best works for them. We could design further levels of the bowling game for further practice etc.
Pencil and Paper: “easier to think things through”?
In questionnaires I have asked students to complete following such activities, some students state that they find it “harder to think” using a computer, rather than pencil and paper. Perhaps this is because without tablet writing technology readily available to make notes, scrawl some draft workings, and the presence of a keyboard that keeps them at a physical distance from the questions and their working (rather than just a thin, flat, screen, as for ipads etc.) it is less easy to engage/interact/work through/reformulate the question. Either way, this suggests that to “remember” some students may prefer a pencil and paper approach. The Equation of Lines resource aims to offer a pencil and paper approach, alongside an insight into why writing an equations for a line (rather than simply drawing one!) may be necessary. The pencil and paper approach is supported by technology to offer instant feedback and clear visual demonstrations of the concept, centred round a structured and scaffolded worksheet that breaks down the problem of an equation of a line into bite size, accessible, chunks. The emphasis is on clear explanation: visually and orally. This may help those students who are relatively conscientious, want to do well, but have very little past experience of success in mathematics and where “experimentation” for them may be synonymous with failure and a loss of time. Of course teachers can create safe and encouraging classroom environments that address this issue, but doing so is often one of the biggest challenges facing practising teachers because it is perhaps indicative of a mastery/competence across a wide range of teaching skills: socially, psychologically and academically.
Applications of Mathematics “how is this useful outside of the classroom?”
Both the above resources can leave the question “how is this useful outside of the abstract world of mathematics?” unanswered. Physical World Sequences aims to make links to the modelling aspects of equations of lines and functions generally. To show students some applications and make links from lines to sequences and arithmetic, reinforcing the number of bonds to which students can attach and integrate this new knowledge. Addressing the Cockroft report's concern: ““You collect like terms, or learn the laws of indices, with no perception of why anyone needs to do such things”. Youtube is a fantastic resource for providing a window to the applications of mathematics ( Bringing the outside world).
The METAL (Mathematics for Economics: Enhancing Teaching and Learning) maths project also has a series of great videos on applications of mathematics in economics.
Centre for Innovation in Mathematics Teaching
Practice is also a necessary part of mastery. Singapore, regularly ranking in the top three in international comparisons of mathematical achievement, have “considerable practice in any mathematical concept covered. This was illustrated by the number of exercises set on each topic, probably about five times the amount set on a similar topic in a British textbook” ( “MEP The First Three Years”, Burghes, D.N., Exeter University, p.4)1. The Centre for Innovation in Mathematics Teaching started by the University of Exeter provides some very useful, free, self-marking exercises often with some clear and interactive explanation preceding the exercises e.g. Plotting Graphs Given their Equations
www.mymaths.co.uk has taken the CIMT model a stage further. Students are able to sit the same homework task as many times as they wish (due to the random generation of questions) and see their results improve. Each task has a detailed, student interactive explanation for revising, or even learning, the topic being covered prior to sitting the practice exercise.
The Khan Academy is a further example of “clear explanations”, “instant feedback” and “practice”. Los Altos educators have signed up for the full Khan Academy package with allows them to keep a track on students progress on different topics studied through Khan Academy videos and practice exercises . The application generates lots of data on the progress of different students right down to which questions caused them problems. The teacher has access to all this data ahead of class and can decide which students to see about what topics.
Khan’s recurring argument behind this “Flipped Classroom” method of instruction is that contrary to perhaps the older generations instinctive view on students interacting with video, flipping the role of lesson time and homework actually “humanises the classroom”. In traditional classrooms “5% of teacher time is spent on one-to-one student help compared to 100% of teacher time spent one-to-one in the flipped classroom”. It encourages dialogue between students and their teacher and perhaps a deeper insight for the teacher about their pupils’ relative difficulties. www.ibvodcasting.com is another good website for self-help mathematics tuition (started by the mathematics department at the Munich International School).
Criticism of the Khan Academy and “Flipped Classroom” style of instruction is that it focuses on “how to do an algorithm correctly” on “calculation” rather than “reasoning, exploration and investigation”. It risks presenting mathematics as a series of algorithms to be mastered: factorising a quadratic, solving equations, converting recurring decimals to fractions. The “Computer Based Math” (www.computerbasedmath.org) initiative and the New Commission on the Skills of the American Workforce: Tough choices suggest that such skills are outdated, perhaps even an obsolete skillset in the technological era, and detrimental to developed countries future long term economic prosperity. In Conrad Wolfram’s view: “it seems completely nuts/backwards to be watching these nice presentations on computer telling you how to solve a quadratic by hand. The student shouldn’t be trying to solve an equation by hand, the computer should be doing that. The question should be, ‘why are you needing to solve this equation? What got set up/modelled into an equation that you needed/wanted to solve and what are you going to do with it when it comes out the other end? The computer is used to replace the teacher rather than to replace calculating, which seems to me to be a mistake. It makes questions very close ended instead of open ended. Let’s lock it down and make something that is especially for school, the student isn’t being encouraged to explore, ask questions, investigate" (Working Through the Objections to CBM: Part 1, as of 12mins30).
How do you assess understanding and problem solving ability ?
Mathematical Olympiads are a fantastic source of questions that test understanding. The question below, taken from the UKMT Junior Challenge in 1997, is a good question for illustrating the concern often voiced in Mathematical education for “rote learning” as opposed to “conceptual understanding”.
A common error of students (and parents!), given the time limit, is to mechanically apply the “rote learnt” method of completing the additions and then converting all fractions to the same common denominator: like cracking a nut with a hammer, but more time consuming. The “conceptual understanding / elegant” solution is to remove those fractions for which numerator < denominator and then realise that the numerator and denominator of the remaining fractions differ by only one unit (spotting patterns), hence the one with the smallest denominator must be the biggest (conceptual understanding of fractions notation). Certainly exposure to these type of questions, prolonged investigations and examples of applications are essential elements of a sound mathematical education. Yet the benefits from practice and mathematical knowledge in achieving understanding, solving problems and identifying productive paths for investigation can be underestimated. My experience at inter-school mathematics competitions e.g. ISMTF, and teaching students from different national education systems (Korean, American, British, German, French, Spanish etc.) suggests that, all other things being equal, greater mathematical knowledge increases the range of options that are stimulated by any given question and increased familiarity/integration of this knowledge increases the speed and flexibility with which these different options can be applied, modified and combined (echoing the Singapore example mentioned above). Memory and/or previous exposure and practice play a significant role in problem solving and selecting fertile paths of investigation (once/provided the student has had at least some exposure to such conceptual questions and prolonged investigations).
Understanding and Memory
It is helpful to have resources to support both understanding and memory. This blog entry from Kris Boulton provides a thoughtful reflection on the interaction between memory, understanding and performance in applying mathematics successfully in unfamiliar contexts.
There are many different routes to understanding. We all arrive in classes of learning with different past experiences, knowledge and interests.
This resource on Dividing Fractions makes the link to multiplying fractions (with a link to a geogebra applet that provides a great conceptual visualisation of fractional multiplication) and offers a very clear explanation, and visual representation, for the types of problem that equations such as 3 ÷ represent. However, for a number of students, having a conceptual grasp for dividing fractions can prove harder than “remembering the rule” for how to do it. Moreover, having a conceptual grasp of fractional division does not necessarily imply being able to quickly divide any two fractions. Some students may find it easier to start with the rule and then later, want to understand why. Others may wish first to understand the concept, and then to gain proficiency in the technique/algorithm. There is not one path to understanding, nor to proficiency. Moreover, whichever path a student takes, memory aids are likely to prove helpful. Videos and song, or a combination of the two (creating “anchors” in more than one sensorial stimulus) can be very effective and is greatly facilitated by modern technology. This Dividing Fractions song from songsofhigherlearning.com has proved a huge success in my classes with a number of students. It has facilitated success in an area where they had previously experienced repeated failure. With this success has come increased motivation and belief that they can do mathematics, producing greater perseverance – all necessary requirements for the sustained effort and genuine engagement required to understand “why”. Other students, once they understood “why”, then had the necessary motivation, belief and perseverance to learn “how” to perform the algorithm. This is a similar/the same point I think Kris Boulton is making in his reflection on how the process of committing a poem to memory can be one of the most effective, self-motivated paths to understanding. This Human Transformations resource is a great activity for introducing the concept of how equations can define geometries other than straight lines. Yet when it comes to analysis, some more detailed practice is needed in understanding the function as defining the relationship between an x-coordinate and its corresponding y-coordinate: Giant Function Transformations. The best is probably a balanced diet of both, but different students are likely to require each element in different proportions.
Exams, Computer Based Learning and Industry Criticism
Conrad Wolfram in his TED talk states that 80% of most Mathematic education systems worldwide focus on calculation/computation. Frustratingly, this is exactly the thing that computers do far more efficiently and effectively than humans. Mr. Wolfram argues that we should be focusing our Mathematics classes on points 1,2 and 4 below and that we can only do that if classes have extra time because they have used the computer to deal with step 3.
1. Posing the right questions
2. Convert Real world problems into a mathematical formulation
3. Computation (80% of time done by hand in current Mathematics curricula)
4. Reapply insights developed from our mathematical models to the real world to test them.
Wolfram demonstrates what a computer based curriculum may look like using an example of projectile motion (Working Through the Objections to CBM: Part 1, as of 19minutes). “It is, under current curriculum categorisations, a quadratics question, but such questions that appear in our current curriculums can all be solved on computer yet the answers are actually totally wrong because we are using an oversimplified model of the actual path of the projectile from a gun. We’re dumbing the question down to make it simple enough to do with pencil and paper. With the time saved by using a computer the student can ask themselves: “what don’t I know?". A lot of the intellectual challenge isn’t doing it, it’s asking: “am I looking at a useful formula, what don’t I know from this formula, where on the web can I look to find out about drag, which leads to the information that it’s dependent on air pressure, that air density goes down as the projectile moves higher, gravity also gets lower and this still doesn’t cover everything that might be in a perfect model like the curvature of the earth, 3D geometry and the Coriolis effect etc. These are the things we don’t teach our students to ask about and look for, when we should be, instead of giving every piece of information they need in the question e.g. if the winds blowing plus or minus ten miles per hour faster what difference will this make to its trajectory?”. Incidentally, information on drag, air pressure etc. were found by looking them up on Wikipedia (ibid, from 20min22), which is within the capabilities of school students just as well as trained mathematicians e.g. Motion of Projectile With or Without Air Resistance, Understanding Kinematics and Newton's Laws of Motion.
The “New Commission on the Skills of the American Workforce” report Tough choices mirrors these concerns: “our testing system rewards students who will be good at routine work, while not providing opportunities for students to display creative and innovative thinking and analysis.”(p.9). “jobs that are most vulnerable are the jobs involving routine work. If someone can figure out the algorithm for a routine job, chances are that it is economical to automate it. Many good, well-paying, middle-class jobs involve routine work of this kind and are rapidly being automated” (p.5). “A world in which routine work is largely done by machines is a world in which mathematical reasoning will be no less important than math facts, in which line workers who cannot contribute to the design of the products they are fabricating may be as obsolete as the last model of that product, in which auto mechanics will have to figure out what to do when the many computers in the cars they are working on do not function as they were designed to function, in which software engineers who are also musicians and artists will have an edge over those who are not as the entertainment industry evolves, in which it will pay architects to know something about nanotechnology, and small business people who build custom yachts and fishing boats will be able to survive only if they quickly learn a lot about the scientific foundations of carbon fibre composites” (p.7).
Different Occupations require different Mathematical skills
I find these recommendations from Mr. Wolfram and the Tough Choices report exciting and pertinent. However, I have difficulty envisaging what such a mathematics curriculum would look like in practice for younger high school students aged 11 through to 14 and those for whom Mathematics does not come easily. I look forward to seeing in more detail the programmes that have been announced for the Estonian probability and statistics module, which I hope will clarify how this translates into educational practice and from where, we as teachers, can begin designing resources along similar lines.
I am also interested in how this curriculum would address the needs of industry cited in the Cockroft report “Mathematics Counts” (p.11 paragraph 39) as: “an 'at-homeness' with numbers and an ability to make use of mathematical skills which enables an individual to cope with the practical mathematical demands of his everyday life. The second is an ability to have some appreciation and understanding of information which is presented in mathematical terms, for instance in graphs, charts or tables or by reference to percentage increase or decrease”. This resource, Comparing Data Distributions, looking at the interpretation of boxplots and cumulative frequency graphs, focuses on these concerns, as do Interpreting World Statistics and AIDS-HIV statistics. These industry requirements appear as true today as they were back in 1982 if one looks at the mathematics sections of modern advanced (post 18) apprenticeships tests. These tests are very much focused on the mathematical skills identified by the Cockroft committee as essential to any adult life, regardless of occupation: “it is important to have the feeling for number which permits sensible estimation and approximation - of the kind, for instance, which makes it possible to realise that the cost of 3 items at 95p each will be a little less than £3 - and which enables straightforward mental calculation to be accomplished” e.g. Percentages of Amounts. An inability to understand ratio, 5% or 10% of a given quantity, mentally estimate the answer to a calculation are skills that, if absent after five years of compulsory secondary education, shock employers and adults alike. Some of the examples I have seen given with computer based maths are very relevant to my post 16 year old mathematics students looking towards engineering, high finance or a career in science and I think are likely to help with the Tough Choices (p.5) conclusion that: “Those countries that produce the most important new products and services can capture a premium in world markets that will enable them to pay high wages to their citizens”. These higher wages then filter through the economy as the new products and services workers, with premium wages, buy further goods and services and pay higher taxes. I’m not sure the interpretation of graphs and Cockroft’s “'at-homeness' with numbers” fall into the same category as Conrad Wolfram’s concerns, but certainly agree that it seems misplaced and less inspiring to undertake statistical analysis and presentation without the use of computers.
Mathematically able adults get excited about certain areas of mathematics because they know where it has taken human knowledge, applications and technological development generally. Most students do not live in an adult world nor do they have adult emotional reactions or experiences. Resources such as those offered at www.numberloving.co.uk, grounded as they are in youth culture and experience, provide useful bridges from the adolescent and pre-adolescent world towards the mathematical skills that will prove insightful in their future adult lives. A variety of well selected Youtube videos can also greatly help bridge this gap and provide a vision and inspiration for where this knowledge can take them [Memory: song - Mr. Duey, Humour - Convert Improper Fractions to Mixed Numbers, Inspiration: Quantum stealth camouflage, Exoskeletons (also being used for people with reduced mobility diseases), Google Cars (Feb 2012, now authorised in certain states in the US (as of July 2013) worth watching the whole video)]. A large part of a teacher’s role is to form a bridge between the pre-adolescent / adolescent world and the adult one. To meet students on their own ground and lead them from there to new ideas, knowledge and possibilities.
I hope this blog offers some examples of the differences between problem solving, investigation/exploration, memory aids, clear explanations, motivation and mathematical applications and how all of these are essential parts of a balanced “mathematical diet”. That memory and understanding are both necessary to effective learning and productive mathematics in problem solving, applications and investigation, even as we move towards, perhaps, a more “computer based” model of mathematical education.
(1) Interestingly schools in Japan and South Korea do not necessarily have more hours of schooling than in the UK, US or Australia [ The Independent newspaper: "English pupils spend around 7,250 hours in the classroom, far more than South Korea, at 6,000, and Japan, at 6,300" - see last paragraph] despite displaying similar increases to Singapore in time spent practising Mathematics. The time required for this additional practice appears to be a cultural situation in which typically students continue to work (either with the help of parents, private tutors or extra group tuition) after school [Centre for Innovation in Mathematics Teaching, University of Exeter, p.4]