Tag: algebraic thinking

Posted in Algebra

How to grow algebra eyes and ears

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Math teachers should grow algebra eyes and ears.  To have algebra eyes and ears means to be always on the lookout for opportunities for students to engage in  algebraic thinking which involves thinking in terms of generality and to reason in terms of relationships and structure, etc. In the post Teaching algebraic thinking without the x’s I described some tips on how to engage pupils in algebraic thinking as they learn about numbers. Likewise in Algebraic thinking and subtracting integers and Properties of Equality – do you need them to solve equation?

Here is another example. How will you use this number patterns in your algebra 1 class so students will also grow algebra eyes and ears?

Let me share how I teach this. I like to simply post this kind of patterns on the blackboard without any instruction. For a few seconds students would normally not do anything and wait for instruction but getting none would start scribbling on their notebooks. When asked what they’re doing they would tell me they are generating other examples to check if the the pattern they see works (yes, detecting patterns is a natural tendency of the mind). When I asked what’s the  pattern and how they are generating the examples I sometimes get this reasoning:  the first and the second columns increase by 1 so the next must be 5 and 6 respectively, the third and fourth columns increase first by 6, then by 8 so the next one must increase by 10 so the next numbers must be 30 and 31 respectively. That is, 5^2 + 6^2 +30^2 = 31^2. Of course this is not what I want so I would ask them if there are other ways of generating examples that does not depend on any of the previous cases.

In generating examples, students usually start with the leftmost number. I would challenge them to start from any terms in the equation. After this, if no one thought of proving that the pattern will work for all cases, then I’ll ask them to prove it. It would be easier for me and for them if I will already write the following equation at the bottom of the pattern for students to fill up and prove but this method is for the lazy and lousy teacher. A good algebra teacher never gives in to this temptation of doing the thinking of representing an unknown by a letter symbol for their students.

In proving the identity, I have observed that students will automatically simplify everything so they end up with fourth degree expressions. This is another opportunity to challenge the students: show that the left hand side and right hand side simplifies to identical second degree expressions with only their knowledge of square of the sum (a+b)^2 = a^2+2ab+b^2.

The teaching sequence I just described is consistent with the levels of understanding of equation I described in Assessing understanding of function in equation form.

proportion
Posted in Teaching mathematics

What is proportional reasoning? Does cross multiplication help learn it?

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proportionProportional reasoning is a capstone of children’s elementary school arithmetic and a cornerstone of all that is to follow (Lesh & Post, 1988). But for some reason, the teaching of elementary school mathematics topics either become an end in itself or has become more a preparation for learning algebra. Proportional reasoning is not being given its due attention. Solving proportion problems has become an exercise of applying routine procedure than an opportunity to engage students in proportional reasoning.

What is proportional reasoning? Why is it important?

Proportional reasoning is a benchmark in students’ mathematical development (De Bock, Van Dooren, Janssens, & Verschaffel, 2002). It is considered a milestone in students’ cognitive development. It involves:

  1. reasoning about the holistic relationship between two rational expressions such as rates, ratios, quotients, and fractions;
  2. synthesis of the various complements of these expressions;
  3. an ability to infer the equality or inequality of pairs or series of such expressions;
  4. the ability to generate successfully missing components regardless of the numerical aspects of the problem situation; and
  5. involves both qualitative and quantitative methods of thought and is very much concerned with prediction and inference.

Proportional reasoning involves a sense of co-variation and of multiple comparisons. In this sense it is a ‘subset’ of algebraic thinking which also give emphasis on structure and thinking in terms of relationship.

What is cross multiplication? Does it promote proportional reasoning?

Cross multiplication is a procedure for solving proportion of the type A/B = x/D. It solves this equation by this process: A*D = x*B. This algorithm is not intuitive. It is not something that one will ‘naturally generate”. Studies have consistently shown that only very few students understand it although many can carry out the procedure. I know many teachers simply tell the students how to do cross multiplication and use specific values to verify that it works without explaining why the algorithm is such.

Many mathematics textbooks and lessons are organized in such a way that students are taught to do cross multiplication before asking them to do problems involving proportion. This practice deprives the students from understanding the idea of proportion and developing their proportional thinking skills. Research studies recommend to defer the introduction of cross multiplication until students have fully understood proportion and have had experiences in solving proportion problems using their knowledge of operation and their understanding of fraction, ratios, and proportion.

References and further readings:

  1. Number Concepts and Operations in the Middle Grades
  2. Proportional reasoning tasks and difficulties
  3. Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning for Teaching Mathematics: Grades 6-8
Posted in Algebra, Curriculum Reform

Teaching algebra – it pays to start early

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I believe in early algebraization. I have posted a few articles in this blog on ways it can be taught in the early grades. Check out for example  Teaching  Algebraic Thinking Without the x’s. All the lessons in fact that I post here whether it is a number or geometry or pre-algebra lesson always aim at developing students’ algebraic thinking. What do research say about early algebraization? How do can we integrate it in the grades without necessarily adding new mathematics content?

“Traditionally, most school mathematics curricula separate the study of arithmetic and algebra—arithmetic being the primary focus of elementary school mathematics and algebra the primary focus of middle and high school mathematics. There is a growing consensus, however, that this separation makes it more difficult for students to learn algebra in the later grades (Kieran 2007). Moreover, based on recent research on learning, there are many obvious and widely accepted reasons for developing algebraic ideas in the earlier grades (Cai and Knuth 2005). The field has gradually reached consensus that students can learn and should be exposed to algebraic ideas as they develop the computational proficiency emphasized in arithmetic. In addition, it is agreed that the means for developing algebraic ideas in earlier grades is not to simply push the traditional secondary school algebra curriculum down into the elementary school mathematics curriculum. Rather, developing algebraic ideas in the earlier grades requires fundamentally reforming how arithmetic should be viewed and taught as well as a better understanding of the various factors that make the transition from arithmetic to algebra difficult for students.

The transition from arithmetic to algebra is difficult for many students, even for those students who are quite proficient in arithmetic, as it often requires them to think in very different ways (Kieran 2007; Kilpatrick et al. 2001). Kieran, for example, suggested the following shifts from thinking arithmetically to thinking algebraically:

  1. A focus on relations and not merely on the calculation of a numerical answer;
  2. A focus on operations as well as their inverses, and on the related idea of doing/undoing;
  3. A focus on both representing and solving a problem rather than on merely solving it;
  4. A focus on both numbers and letters, rather than on numbers alone; and
  5. A refocusing of the meaning of the equal sign from a signifier to calculate to a symbol that denotes an equivalence relationship between quantities.
These five shifts certainly fall within the domain of arithmetic, yet, they also represent a movement toward developing ideas fundamental to the study of algebra. Thus, in this view, the boundary between arithmetic and algebra is not as distinct as often is believed to be the case.
What is algebraic thinking in earlier grades then? Algebraic thinking in earlier grades should go beyond mastery of arithmetic and computational fluency to attend to the deeper underlying structure of mathematics. The development of algebraic thinking in the earlier grades requires the development of particular ways of thinking, including analyzing relationships between quantities, noticing structure, studying change, generalizing, problem solving, modeling, justifying, proving, and predicting. That is, early algebra learning develops not only new tools to understand mathematical relationships, but also new habits of mind.”

The foregoing paragraphs were from the book Early Algebraization edited by Jinfa Cai and Eric Knuth. The book is a must read for all those doing or intending to do research about teaching algebra in the elementary grades. Educators and textbook writers should also find a wealth of ideas on how algebra can be taught and integrated in the early years. Of course it would be a great read for teachers.  The book is rather expensive but if you have the money, why not? Here are some section titles:
  • Functional thinking as a route in algebra in the elementary grades
  • Developing algebraic thinking in the early grades: Lessons from China and Singapore
  • Developing algebraic thinking in the context of arithmetic
  • Algebraic thinking with and without algebraic representation: A pathway to learning
  • Year 2 to 6 students’ ability to generalize: Models, representations, and theory for teaching and learning
  • Middle school students’ understanding of core algebraic concepts: equivalence & variable”

Check out the table of contents for more.

The following books also provide excellent materials for developing algebraic thinking.

 

 

 

 

Please share this post to those you think might find this helpful.

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.