Tag: conceptual understanding

Posted in Mathematics education

NCTM Process Standards vs CCSS Mathematical Practices

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The NCTM process standards, Adding it Up mathematical proficiency strands, and Common Core State Standards for mathematical practices are all saying the same thing but why do I get the feeling that the Mathematical Practices Standards is out to get the math teachers.

The NCTM’s process standards of problem solving, reasoning and proof, communication, representation, and connections describe for me the nature of mathematics. They are not easy to understand especially when you think that school mathematics is about stuffing students with knowledge of content of mathematics. But, over time you find yourselves slowly shifting towards structuring your teaching in a way that students will understand and appreciate the nature of mathematics.

The five strands of proficiency were also a great help to me as a teacher/ teacher-trainer because it gave me the vocabulary to describe what is important to focus on in teaching mathematics.

With the Mathematical Practices Standards I had this picture of myself in the classroom with a checklist of the standards in one hand and a lens on the other looking for evidence of proficiency. The NCTM and Adding it Up standards actually said more about math. The ones in Common Core are saying more about what students should attain. I wonder which will encourage ‘teaching to the test’. The day teachers start to ‘teach to the test’ is the beginning of the end of any education reform.

NCTM Process Standards

Five Strands of Mathematical Proficiency

CCSS Mathematical Practices

Problem Solving

  1. Build new mathematical knowledge through open-ended questions and more-extended exploration;
  2. Allow students to recognize and choose a variety of appropriate strategies to solve problems;
  3. Allow students to reflect on their own and other strategies for solving problems.

Reasoning and Proof

  1. Recognize and create conjectures based on patterns they observe;
  2. Investigate math conjectures and prove that in all cases they are true or that one counterexample shows that it is not true;
  3. Explain and justify their solutions.

Communication:

  1. Organize and consolidate their mathematical thinking in written and verbal communication;
  2. Communicate their mathematical thinking clearly to peers, teachers, and others;
  3. Use mathematical vocabulary to express mathematical ideas precisely.

Connections

  1. Understand that mathematical ideas are interconnected and that they build and support each other;
  2. Recognize and apply connections to other contents;
  3. Solve real world problems with mathematical connections.

Representation

  1. Emphasize a variety of mathematical representations including written descriptions, diagrams, equations, graphs, pictures, and tables;
  2. Select, apply, and translate among mathematical representations to solve problems;
  3. Use mathematics to model real-life problem situations.

Conceptual Understanding refers to the “integrated and functional grasp of mathematical ideas”, which “enables them [students]
to learn new ideas by connecting those ideas to what they already know.”

Procedural fluency is defined as the skill in carrying out procedures flexibly, accurately, efficiently, and
appropriately.

Strategic competence is the ability to formulate, represent, and solve mathematical problems.

Adaptive reasoning is the capacity for logical thought, reflection, explanation, and justification.

Productive disposition is the
inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

Mathematically proficient students …

  • Make sense of problems and persevere in solving them.
  • Reason abstractly and quantitatively.
  • Construct viable arguments and critique the reasoning of others.
  • Model with mathematics.
  • Use appropriate tools strategically.
  • Attend to precision.
  • Look for and make use of structure.
  • Look for and express regularity in repeated reasoning.

Image from 123RF

Posted in Mathematics education, Teaching mathematics

Three Levels of Math Teachers Expertise

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Level 1 – Teaching by telling

The teachers at Level 1 can only tell students the important basic ideas of mathematics such as facts, concepts, and procedures. These teachers are more likely to teach by telling. For example in teaching students about the set of integers they start by defining what integers are and then give students examples of these numbers. They give them the rules for performing operations on these numbers and then provide students exercises for mastery of skills. I’m not sure if they wonder later why students forget what they learn after a couple of days.

Levels of teaching

Level 2 – Teaching by explaining

Math teachers at Level 2 can explain the meanings and reasons of the important ideas of mathematics in order for students to understand them. For example, in explaining the existence of negative numbers, teachers at this level can think of the different situations where these numbers are useful. They can use models like the number line to show how negative numbers and the whole numbers are related. They can show also how the operations are performed either using the number patterns or through the jar model using the + and – counters or some other method. However these teachers are still more likely to do the demonstrating and the one to do the explaining why a particular procedure is such and why it works. The students are still passive recipients of the teachers expert knowledge.

Level 3 – Teaching based on students’ independent work

At the third and highest level are teachers who can provide students opportunities to understand the basic ideas, and support their learning so that the students become independent learners. Teachers at this level have high respect and expectation of their students ability. These teachers can design tasks that would engage students in making sense of mathematics and reasoning with mathematics. They know how to support problem solving activity without necessarily doing the solving of the problems for their students.

The big difference between the teacher at Level 2 and teachers at Level 3 is the the extent of use of students’ ideas and thinking in the development of the lesson. Teachers at level 3 can draw out students ideas and use it in the lesson. If you want to know more about teacher knowledge read Categories of teacher’s knowledge. You can also check out the math lessons in this blog for sample. They are not perfect but they are good enough sample. Warning: a good lesson plan is important but equally important is the way the teacher will facilitate the lesson.

Mathematical Proficiency

The goal of mathematics instruction is to help students become proficient in mathematics. The National Research Council defines ‘mathematical proficiency’ to be made up of the following intertwined strands:

  1. Conceptual understanding – comprehension of mathematical concepts, operations, and relations
  2. Procedure fluency – skill in carrying out procedure flexibly, accurately, efficiently, and appropriately
  3. Strategic competence – ability to formulate, represent, and solve mathematical problems
  4. Adaptive reasoning – capacity for logical thought , reflection, explanation, and justification
  5. Productive disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. (NRC, 2001, p.5)

I think it will be very hard to achieve these proficiencies if teachers will not be supported to attain Level 3 teaching I described above. No one graduates from a teacher-training institution with a Level 3 expertise. One of the professional development teachers can engage to upgrade and update themselves is lesson study. The  book by Catherine Lewis will be a good guide: Lesson Study: Step by step guide to improving instruction.

Posted in Algebra

Why negative times a negative is positive

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Among the ‘rules’ for working with negative numbers,  the most counter intuitive is “negative times a negative is a positive”. It is easily forgotten especially if it was learned by rote. It is also not an easy ‘rule’ to make sense of so it needs to be learned with conceptual understanding. Here’s my proposed lesson for teaching multiplication of integers. This lesson takes from the lesson Subtracting integers using tables- Part 1 and Algebraic thinking and subtracting integers – Part 2. Note that this lesson like the rest of the lessons in this blog is not just about students learning the math but more about them engaging in mathematical thinking processes such as searching for patterns, making generalization, reasoning, making connections, etc.

Set the task

Fill up this table  by multiplying the numbers in the first column to the number in the first row. Start filling up the rows or columns you think would be easier to do.

For discussion purposes divide the table into 4 quadrants. The top right quadrant is Quadrant 1, top left is Quadrant 2, bottom left is Quadrant 3, and bottom right is Quadrant 4. This is also one way of leading the students to consider filling-up the quadrants according to their number label.

Explore, Observe, Explain why

Students are more likely to fill-up Quadrant 1 because the numbers to be multiplied are both positive. The next quadrant they are more likely to fill-up is Quadrant 2 or 4. You may want to give the following questions to scaffold their thinking: What do you observe about the row of numbers in Quadrant 1? How can it help you fill up quadrant 2? Do the numbers you put in Quadrant 2 make sense? What does 3 x -2 mean? What about in Quadrant 4? 

From Quadrant 2 students are more likely to fill up Quadrant 3 or Quadrant 4 by invoking the pattern. Questions for discussion:   Do the numbers in Quadrant 4 make sense? What does -3 x 2 mean? This is one way of making the students be aware that commutativity holds in the set of integers. The problematic part are the numbers in Quadrant 3. None of the previous arguments are useful to justify why negative times negative is positive except by following the patterns. But this explanation will be enough for most students. You can also use the explanation below.

Revisit the rule when teaching another topic

We know that 8 x 8 = 64. This means that (10-2)(10-2)=64.  By distributive property, (10-2)(10-2)= 100+-2(10) + – 2(10)+ ____ = 64. Previously students learned that -2(10)= 20. Hence, 100 + -40+___= 64. What should go in the blank must be 4. So (-2)(-2) = 4. This proof was first actually proposed by Maestro Dardi of Pisa in year 1334. In explaining this to students I suggest rewriting (10-2)(10-2) to (10+-2)(10+-2) to reinforce the distinction between the dash sign as minus and as symbol denoting ‘negative’.

Girolamo Cardano sometime in 1545 proposed a geometric interpretation of this operation. He argued that (10-2)(10-2) can be interpreted as cutting off 2 strips of 2 x 10 rectangles from the two sides of the 10 x 10 square. Cutting the rectangles like these meant cutting the  2 x 2 square twice so you need to return back the other square. The figure below shows this. This proof by Cardano is usually used to teach the identity square of a difference (x-y)(x-y)=x^2-2xy+y^2. This is a good opportunity to revisit the rule negative times a negative is positive.

Reference to history is from the paper Historical objection to the number line by Albrecht Heefer.

Posted in Algebra, High school mathematics

Using cognitive conflict to teach solving inequalities

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One way to teach and assess students understanding of math concepts and procedures is to create a cognitive conflict. Here is one way you can create cognitive conflict in solving inequalities:

To solve the inequality x – 7 > 5, the process usually involve adding 7 to both sides of the inequality.

solving_inequality

This process uses the principle a > b then a + c > b + c. There is no change in the inequality sign since the same number is added to both side.

Now, what if we add 7 to the left side of the inequality and 6 to the right side?

cognitive conflict

The process uses this principle: If a > b, c > d then a + c > b + d. Should this create a change in the inequality sign? Certainly not. There should be no change in the inequality sign when a bigger (smaller) number is added to the bigger (smaller) number side.  Both of these processes create a cognitive conflict and will be a good opportunity for your class to discuss what solving inequality means and, compare the processes of solving equations and inequalities. Comparing and contrasting procedures is also a good strategy to developing conceptual understanding.

For those interested to learn more about inequalities I recommend this book:Introduction to Inequalities (New Mathematical Library)