Posted in Algebra, Curriculum Reform, Elementary School Math, What is mathematics

“New Math” curriculum

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math curriculumIn the video below, Tom Lehrer ‘explains’ and ‘criticizes’ the New Math curriculum in a funny way. I think the New Math curriculum was not that bad really. It helped the teachers to structure their math knowledge in a “mathematical” way. This is a good thing. The trouble was they taught mathematics in the mathematical way. Enjoy the video.

What is the New Math Curriculum?

New Mathematics or New Math was a brief, dramatic change in the way mathematics was taught in American grade schools, and to a lesser extent in European countries, during the 1960s. The name is commonly given to a set of teaching practices introduced in the U.S. shortly after the Sputnik crisis in order to boost science education and mathematical skill in the population so that the intellectual threat of Soviet engineers, reputedly highly skilled mathematicians, could be met.

New Math emphasized mathematical structure through abstract concepts like set theory and number bases other than 10. Beginning in the early 1960s the new educational doctrine was installed, not only in the USA, but all over the developed world.

Criticisms of the New Math Curriculum

Parents and teachers who opposed the New Math in the U.S. complained that the new curriculum was too far outside of students’ ordinary experience and was not worth taking time away from more traditional topics, such as arithmetic. The material also put new demands on teachers, many of whom were required to teach material they did not fully understand. Parents were concerned that they did not understand what their children were learning and could not help them with their studies. In the end it was concluded that the experiment was not working, and New Math fell out of favor before the end of the decade, though it continued to be taught for years thereafter in some school districts.

In the Algebra preface of his book “Precalculus Mathematics in a Nutshell,” Professor George F. Simmons wrote that the New Math produced students who had “heard of the commutative law, but did not know the multiplication table.”

— excerpts taken from Wikipedia

Posted in Mathematics education

How confident are you to teach mathematics?

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As mathematics teachers we simply cannot just stop learning and improving in our field. Reflecting on our practice is a powerful and productive way of supporting our own professional development. I found a goldmine of tools for this in the National Center for Excellence in the Teaching Mathematics (NCETM). I think this site is great for mathematics teachers who wants to keep on improving their craft. Below are some of the self-evaluation questions they have for mathematics- specific teaching strategies.

1. How confident are you that you know how and when it is appropriate to:

  • demonstrate, model and explain mathematical ideas?
  • use whole class discussion?
  • use open questions with more than one possible answer to challenge pupils and encourage them to think?
  • use higher order or more demanding questions to encourage pupils to explain, analyse and synthesise?
  • intervene in the independent work of an individual or group?
  • summarise and review the learning points in a lesson or sequence of lessons?

2. How confident are you that you can select activities for pupils that will promote your learning aims and, over time, give them opportunities to:

  • work independently as individuals or collaboratively with others?
  • engage in interesting and worthwhile mathematical activities?
  • investigate and ‘discover’ mathematics for themselves?
  • make decisions for themselves?
  • reason and develop convincing arguments?
  • visualise?
  • practise techniques and skills and remember facts in varied ways and contexts?
  • engage in peer group discussion?
  • communicate their results, methods and conclusions to different audiences?
  • appreciate the rich historical and cultural roots of mathematics?
  • understand that mathematics is used as a tool in many different contexts?

3. How confident are you that you know how and when you might provide:

  • alternative or supplementary activities for pupils who experience minor difficulties with learning?
  • mathematical activities designed to respond to pupils’ diverse learning needs, including special educational needs?
  • suitable activities for mathematically gifted pupils?
  • suitable homework?

4. How confident are you that you are familiar with a range of equipment and practical resources to support mathematics teaching and learning, such as:

  • structural apparatus and other models for teaching number?
  • measuring equipment?
  • resources to support the teaching of geometrical ideas?
  • board games and puzzles?
  • resources to support and stimulate data handling activities?
  • calculators?
  • ICT and relevant software?

Here are sample questions for self evaluation about mathematics content knowledge. Go to the NCETM.org site for other topics.

1. How confident are you that you know and can explain the properties of:

  • the sine function?
  • the cosine function?
  • the tangent function?

2. How confident are you that you can explain:

  • why sin ? / cos ? = tan ? and use this to solve simple trigonometric equations?
  • why sin² ? + cos² ? = 1 and use this to solve simple trigonometric equations?

Please share this with your co-teachers.

 

Posted in Algebra, High school mathematics, What is mathematics

What is mathematical modeling?

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While there is no consensus yet as to a precise definition of this term, mathematical modeling is generally understood as the process of applying mathematics to a real world problem with a view of understanding the latter. One can argue that mathematical modeling is the same as applying mathematics where we also  start with a real world problem, we apply the necessary mathematics, but after having found the solution we no longer think about the initial problem except perhaps to check if our answer makes sense. This is not the case with mathematical modeling where the use of mathematics is more for understanding the real world problem. The modeling process may or may not result to solving the problem entirely but  it will shed light to the situation under investigation. The figure below shows key steps in modeling process.

Mathematical modeling approaches can be categorized into four broad approaches: Empirical models,  simulation models, deterministic models, and stochastic models. The first three models can very much be integrated in teaching high school mathematics. The last will need a little stretching.

Empirical modeling involves examining data related to the problem with a view of formulating or constructing a mathematical relationship between the variables in the problem using the available data.

Simulation modeling involve the use of a computer program or some technological tool to generate a scenario based on a set of rules. These rules arise from an interpretation of how a certain process is supposed to evolve or progress.

Deterministic modeling in general involve the use of equation or set of equations to model or predict the outcome of an event or the value of a quantity.

Stochastic modeling takes deterministic modeling one further step. In stochastic models, randomness and probabilities of events happening are taken into account when the equations are formulated. The reason behind this is the fact that events take place with some probability rather than with certainty. This kind of modeling is very popular in business and marketing.
Examples of mathematical modeling can be found in almost every episode of the TV hit drama series The Numbers Behind NUMB3RS: Solving Crime with Mathematics

The series depicts how the confluence of police work and mathematics provides unexpected revelations and answers to perplexing criminal questions. The mathematical models used may be way beyond K-12 syllabus but not the mathematical reasoning and thinking involve. As the introduction in each episode of Numbers says:

We all use math every day;
to predict weather, to tell time, to handle money.
Math is more than formulas or equations;
it’s logic, it’s rationality,
it’s using your mind to solve the biggest mysteries we know.

The Mathematics Department of Cornell University developed materials on the mathematics behind each of the episodes of the series. You can find the math activities in each episode here.

The challenge in mathematical modelling is “. . . not to produce the most comprehensive descriptive model but to produce the simplest possible model that incorporates the major features of the phenomenon of interest.” -Howard Emmons