Tag: teaching for understanding

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 Elementary School Math, Number Sense

How to scaffold algebraic thinking in teaching integers

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One way to make algebra make sense to students is to show where those mathematics objects (e.g. algebraic expressions/equations/formula) come from. It will even makes sense more to students if they themselves can generate those objects. As John Mason puts it,

The mechanics of algebra (algebraic manipulation) are concerned with studying the effects of combining, undoing and otherwise relating expressions. These make little or no sense unless learners have themselves gained facility in generating expressions so that they know how they arise.  – from Actions and Objects by John Mason.

Generating expressions helps develop algebraic thinking. There are many ways of of embedding this in your teaching. For example with equations, you can ask the students to find as many equations given a solution. Read the post on how to teach the properties of equality on how this can be done. My post about teaching algebraic expressions also shows an example of a task that generates several equivalent algebraic expressions from the same problem situation. Generating formulas by deriving it from other expressions can also be a good activity.  Examples of these is deriving the formula of the area of the triangle from parallelogram/ rectangle and then from these deriving the area of trapezoids. These type of activities help develop students algebraic thinking skills.

Now, how can we do this generating expressions in earlier grades? Let me describe a lesson I taught to a class of year 6 students. This lesson is a continuation of the lesson on teaching integers via the number line with a twist. In that lesson, instead of asking student to arrange numbers, I asked them to arrange number expressions. From there we were able to extend the numbers they know (whole numbers) to now include the negative numbers. The main aim of that lesson is to extend the students’ concept image of negative number from a number that can be used to represent situation (see post on a problem solving approach for introducing integers) to a number that results when you take away a bigger number from a smaller number.

In this lesson with negative numbers, zero, and positive numbers on the number line, I can now proceed to defining integers or perhaps compare integers. But what will the students learn from that except that they are called integers? If I ask them to compare the numbers what good is that at this point? Where will they use that knowledge? So the task that I gave  them in this lesson was to make as many number expressions whose answer corresponds to the numbers in the number line. Note that this task is an open-ended problem solving task. Below is a sample student solution.

When I asked the class to share their answers I was surprised that they did not restrict themselves to addition and subtraction operation.

With the data shown on the board (pardon my handwriting) I asked them to make some observations and generalizations. They gave the following:

1. It is easy to make number expressions when the answer should be positive.

2. You always get a negative if you subtract a bigger number from a smaller number.

3. You always get a zero if you subtract equal numbers. (If you think this knowledge easily transfer to negatives, you’re wrong. I did try my luck when I asked them “is it also true to -7 – (-7)?”. I got blank stares. Clearly the expression it is still beyond them.

If these are not powerful mathematics to you especially #2 and well, #3, I don’t know what they are. Anyway, the point of my story here is that it is good practice to ask students to generate expressions. It is like asking them to think of a problem given the solution, a highly recommended  mathematics teaching practice.

Posted in Curriculum Reform, Mathematics education

Understanding by Design from WikiPilipinas

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I think the following entry from WikiPilipinas needs revising. “Learning of facts”? Check also the last statement.

“Teaching for understanding” is the main tenet of UbD. In this framework, course design, teacher and student attitudes, and the classroom learning environment are factors not just in the learning of facts but also in the attainment of an “understanding” of those facts, such as the application of these facts in the context of the real world or the development of an individual’s insight regarding these facts. This understanding is reached through the formulation of a “big idea”– a central idea that holds all the facts together and makes these connected facts worth knowing. After getting to the “big idea,” students can proceed to an “understanding” or to answer an “essential question” beyond the lessons taught.

One of my initial concerns about UbD in my previous post is about not checking first if the bandwagon we jumped in to will run in our roads although  I received a comment that said the DepEd did pilot it and are confident that it can. The results of the pilot I believe are not for public consumption. We just have to believe their word for it. But with this post at WikiPilipinas, I don’t know if it is clear to us what the wagon is.  Here’s the next paragraph:

Through a coherent curriculum design and distinctions between “big ideas” and “essential questions,” the students should be able to describe the goals and performance requirements of the class. To facilitate student understanding, teachers must explain the “big ideas” and “essential questions” as well as the requirements and evaluative criteria at the start of the class. The classroom environment should also encourage students to work hard to understand the “big ideas” by having an atmosphere of respect for every student idea, including concrete manifestations such as displaying excellent examples of student work.

But I love the description of traditional method of constructing the curricula in the following paragraph. Very honest. But I can’t agree about the analogy with Polya’s.

The UbD concept of “teaching for understanding” is best exemplified by the concept of backward design, wherein curricula are based on a desired result–an “understanding” or a “big idea”–rather than the traditional method of constructing the curricula, focusing on the “facts” and hoping that an “understanding” will follow. Backward design as a problem-solving strategy can even be traced back to the ancient Greeks. In his book “How to Solve It” (1945), the Hungarian mathematician George Polya noted that the Greeks used the strategy of “thinking backward” by knowing what you want as a solution in order to solve a problem.

If I remember right, G. Polya wrote “look back” as the last step for solving a problem. It means you reflect on your solution and answer in relation to the problem. But wait, there is a problem solving strategy called “working backwards” which is probably what is meant here but as an analogy to backward design? Uhmmm …

Oh, by the way, “backward design” is a problem solving strategy?

Not that I’m happy we’re adapting Understanding by Design but who cares if I’m happy with it or not. There isn’t anything I can do in that department but just to help now to make sure we make the most of it. It is is a multimillion peso project. That’s our taxes. The one in WikiPilipinas is by far the only resource in the net for UbD Philippines. If you happen to know other related sites, please share.

Here’s one research about UbD in Singapore. Here’s my other UbD related post