Here’s 5 things that are important to know if you teach math …

**Mathematical objects or learning objects (i.e. using manipulatives or models)**

These help students figure out and explain their thinking. Manipulatives (concrete or virtual) tend to draw out students’ need to explain and focus on different representations and meanings of mathematics concepts and models. In addition, learning objects/manipulatives can actually act as models of understanding (Tichá & Hošpesová, 2009). Objects are especially valuable for students still functioning at a concrete level of thinking. Helping students chose appropriate manipulatives is important as protractors are not usually used to measure straight lines.

**Connectedness (i.e. to real life and to all strands of math)**

Teachers need to make connections between and among different math strands as well as concepts and procedures. Math concepts are interrelated for example, multiplication is repeated addition and subtraction is the opposite of addition. Division and multiplication are interrelated and opposite and are interrelated to fractions, decimals, and ratios. Teachers need to make these connections to prevent students from using math concepts and procedures in isolation. Further, teachers need to connect math to real world applications such as How do carpenters make sure a door is installed right? … they measure the diagonals to make sure they are equal (equal diagonals means the door is installed at right angles). The teaching of math is not presented as a “unified body of knowledge” when taught in singular isolation (Ma,1999, p.122).

**Multiple Perspectives (i.e. solving problems different/flexible ways)**

Teachers need to stress the idea that multiple solutions are possible but also explain that some approaches to solutions and methods are more appropriate in certain situations. This multiple perspective allows students to be flexible in their thinking and understanding of the content.

**Basic Ideas (i.e. key ideas/understandings)**

When teaching math, teachers stress basic ideas and key understandings. For example, when solving a problem, students can use an equation to provide proof of their answer. Showing their answers different ways can reaffirm their proof.

**Longitudinal Coherence (also known as curriculum and learning trajectories i.e. how curriculum is related between grade levels)**

Teachers need to be aware of what is being taught at all levels of the elementary math curriculum and not just the grades that they are teaching or have taught. When teachers know the math curriculum well, they know where their students’ learning has come from and where it is going. When only knowing the assigned math grade level being taught, teachers miss out in identifying students’ gaps in their math learning. When there is a gap in learning a math concept, teachers can employ “numeracy recovery” just as “reading recovery” is used to help struggling readers. When teachers take opportunities to review key understandings, they can put in place the appropriate foundation for students’ future math achievement.

An effective way of presenting this knowledge is following the development of a specific math concept through the grades – see Grade 1 to 6 Multiplication Learning Trajectory below.

Multiplication Learning Trajectories with curriculum

References:

https://buildingmathematicians.wordpress.com/tag/teaching-mathematics/

https://www.mathrecovery.org/pdfs/how-it-works/Math-Recovery-Research-White-Paper.pdf

Ma, L. (1999). *Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States*. Mahwah, NJ: Lawrence Erlbaum Associates.

Tichá, M., & Hošpesová, A. (2009, January). Problem posing and development of pedagogical content knowledge in pre-service teacher training. In *meeting of CERME* (Vol. 6). From proceedings of CERME 6, January 28th-February 1st 2009, Lyon France INRP2010 1