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Posted in Elementary School Math, High school mathematics, Number Sense

Teaching positive and negative numbers

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A popular approach for teaching numbers is to use it to describe a property of an object or a set of object. For example, numbers are used to describe the amount or quantity of fruits in a basket.

In introducing integers, teachers and textbooks presents integers as a set of numbers that can be used to describe both the quantity and quality of an object or idea. Contexts involving opposites are very popular situations to show the uses and importance of positive and negative numbers and the meaning of its symbols. For example, a teacher can tell the class that +5 represents going 5 floors up and -5 represents going five floors down from an initial position.

Mathematics is a language and a way of thinking and should therefore be experienced by students as such. As a language, math is presented as having its own set of symbols and “grammar” much like our spoken and written languages that we use to describe a thing, an experience or an idea.But apart from being a language, mathematics is also a way of thinking. The only way for students to learn how to think is for them to engage them in it!  Here’s my proposed activity for teaching positive and negative numbers that engages students in higher-level thinking as well.

Sort the following situations according to some categories

  1. 3o below zero
  2. 52 m below sea level
  3. $1000 net gain
  4. $5000 withdrawal from ATM machine
  5. $1000 deposit in savings account
  6. 3 kg weight loss
  7. 2 kg weight gain
  8. 80 m above sea level
  9. 37o above zero
  10. $2000 net loss

The task may seem like an ordinary sorting task but notice that the categories are not given. Students have to make their own way of grouping the situations. They can only do this after analyzing each situation, noting commonalities and differences.

Possible solutions:

1.  Distance vs money (some students may consider the reading the thermometer under distance since its about the “length” of mercury from the “base”)

2. Based on type of quantities: amount of money, temperature, mass, length

3. Based on contrasting sense: weight gain vs weight loss, above zero vs below zero, etc.

The last solution is what you want. With very little help you can guide students to come-up with the solution below.

Of course, one may wonder why make the students go through all these. Why not just tell them? Why not give the categories? Well,  mathematics is not in the curriculum because we want students to just learn mathematics. More importantly, we want our students to think critically and creatively hence we need to give them learning experiences that develops good thinking habits. Mathematics is a very good context for learning these.

Here are my other posts about integers:

Posted in Assessment

Teach and assess for conceptual understanding

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To teach for conceptual understanding requires assessing for conceptual understanding. The principles that guide teaching for understanding must be the same principles that should guide assessment. What are some of these?

  • Open-ended, problem solving tasks
    To teach for conceptual understanding, it is not enough that students engage in problem solving task. The tasks should be (1) open-ended which means that it can be solved in many ways using a range of concepts; (2) accessible, that is , not too easy or too difficult but just beyond the students ability; (3) can be extended by changing conditions in the problem so that it can be used for building concepts and for making synthesis and generalization; and, (4) the task should encourage creativity in the problem solver. These, together with right amount of scaffolding from the teacher and assessment tasks possessing the same characteristics is a perfect recipe for understanding mathematics conceptually.
  • Activities that promote mathematical communication
    Mathematics is a language that enables us to communicate ideas with conciseness, clarity and precision both in oral or written form. Students learning experiences should always aim at developing this capacity. They should be given opportunities to talk about mathematics, to speak mathematics, and communicate mathematically through its written symbols.  These are possible with the right mix of collaborative and individual work. Click this link for sample. This also implies that assessment should focus not only on the knowledge the students are acquiring but also on their skill on communicating this knowledge.
  • Tasks that build on students’ previous knowledge
    Teaching should build on the knowledge that students already have. This does not mean simply putting something on top of what they know. Knowledge has to be connected with other knowledge from within and from without. The more connections there are, the more robust is the understanding. Conducting formative assessment can provide teacher with information on how to structure the lesson to help students make connections. Another strategy which I highly recommend is to teach via problem solving. Click here for sample lesson.
  • Discussions that respects reason
    Mathematics is a way of thinking logically and methodically. As such, classroom culture that respects reason must be created both in the teaching and in assessing. Group or whole class discussion and assessment rubrics should give appropriate feedback to the students as to the way they reason and build on each others reasoning or on each others opinion.
Posted in Misconceptions, Number Sense

From whole numbers to integers – so many things to “unlearn”

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A lot of online resources on integers are about operations on integers especially addition and subtraction.  Most of these resources  show visual representations of integer operations. These representations are almost always in the form of jumping bunnies, kitties, frogs, …  practically anything that can or cannot jump are made to jump on the number line. Sometimes I wonder where and when in their math life will the students ever encounter or use jumping on the number line again.  If you want to know why I think number line might not work for teaching operations, click link –  Subtracting  integers using number line – why it doesn’t help the learning.

Of course there may be other culprits apart from rote learning and the numberline model. Maybe there are other things that blocks students’ understanding of integers especially doing operations with them.

Before integers, students’  life with numbers had been all about whole numbers and some friendly fractions and decimals. So it is not surprising that they would have made some generalizations related to whole numbers with or without teachers help. I pray of course that teachers will have no hand in arriving at these generalizations and that if indeed students will come to these conclusions, it should be by the natural course of things.  Here are some dangerous generalizations.

over-generalizations about whole numbers

These generalizations are very difficult to unlearn (accommodate according to Piaget) because based on students experiences they all work and are all true. Now, here comes integers turning all of these upside down, creating cognitive conflict. In the set of integers,

  1. when a number is added to another number it could get smaller (5 + -3 gives 2; 2 is smaller than 5)
  2. the sum of any two numbers can be smaller than both of the addends (-3 + -2 gives -5; -5 is smaller than -3 and -2)
  3. when a number is taken a way from another number, it could get bigger (3 – -2 = 5, 3 just got bigger by 2)
  4. you can get an answer for taking away a bigger number from a smaller number (3 – 5 = -2)
  5. when a number is multiplied by another number, it could get smaller (-3 x 2 = -5)
  6. when a number is divided by another number, it could get bigger (-15/-3 = 5)

On top of these, mathematics is taught as something that gives absolute result. So how come things change?

You may be interested to read my article on Math War over Multiplication. It’s also about overgeneralization.

Feel free to share your thoughts about these.

Posted in Curriculum Reform

Understanding by Design, one more go

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I have so far written three posts about understanding by design. The first is about my  issues about DepEd’s adoption of understanding by design (UbD), the second is about the information posted about UbD Philippines in WikiPilipinas and the third is about curriculum change and UbD. These posts are very popular especially for readers from the Philippines. This is understandable as our Department of Education wants teachers to implement UbD this June 2010, barely two months from now. I don’t know if there’s a training out there about UbD for our public school teachers. Maybe they will have one, a week before the school year starts this June.

Is this backward or forward design?

Anyway, I am writing this post because some readers land on this blog searching for things like “how to teach algebra using UbD”, “teaching integers the UbD way”, etc. I don’t know if they are just looking for lesson plans using UbD which they will never find in this blog or there’s a misconception out there that UbD is a way of teaching. It is not. It is more a way of planning your lesson rather than how to teach your lesson. In fact the only difference that I see between UbD and the current way of planning the lesson is in the format, not in the way you will actually teach the lesson. UbD says theirs uses backward design. In this model you start with thinking on how you will assess understanding before selecting and organizing your learning activities.  For lack of term, let’s call the traditional method forward design. In this model you think about how you will assess understanding after selecting and organizing your learning activities. In both models of course you start with your learning goals. In UbD it’s called enduring understanding, in the traditional one it is called objectives.

I attended an international conference on science and mathematics teaching a few months ago. One of the parallel session presenter reported her research which compares the use of UbD way of planning the lesson and their so called usual way of planning the lesson for science. She said the class taught using UbD performed better than the one taught using the traditional one. So I asked why is that? She said that it’s because the class taught using UbD used inquiry-based teaching and the class taught using the traditional lesson plan format was taught by lecture method. So I asked further: In your country’s traditional way of planning the lesson, is it not possible to organize the lesson using inquiry-based teaching and teach it that way. She said, “of course we can, and we do. It depends upon the teacher”. There you go. Backward or forward design,  it’s still the teaching and not the format nor the way the lesson plan is prepared that spells the difference in learning.