Brain research indicates that the brain learns first at a concrete level (hands-on manipulatives), then at a pictorial level and finally at an abstract level (Souza, 2008). Furthermore, we also know the more senses that are involved in learning, the more powerful the learning becomes.
Differing Learning Styles
So when we use manipulatives, students get to see, hear and touch the math. There are also such a variety of learning styles that if we are not using manipulatives, we are ignoring all those hands-on learners (and visual learners if we are not using pictures or models). Piaget and other brain development researchers agree that from ages 7-12 students are still in the concrete phase of learning. It is when a child reaches 12 years and older that their brains are able to think and learn more abstractly.
Now, we know that these ages vary depending on each child, however, think about the implications of teaching math to children in grades 1-7 without using concrete materials. We have seen the implications; many students never develop number sense and math becomes a hated subject that is nonsensical to them. This affects: their confidence, their ability to learn new ideas in math, their future opportunities both in high school and in higher education, and future job prospects. The research is compelling enough that our curriculum, along with the curricula of most first world countries indicate that students are to be understanding number concepts concretely (using manipulatives), pictorially and symbolically.
Skills Needed in the Workforce
We also need to understand that the skills that our students need as they progress through the grades and into the “real world” are not computational skills! What do they need? Number sense, mathematical reasoning and problem solving, innovation, creativity, and collaboration. I often hear high school teachers say “they need their basic facts!” I counter with this response “they need basic number sense!”. These are not the same thing.
Basic facts are the memorized facts of single digit addition, subtraction, multiplication and division. It is obviously useful for students to be able to recall these facts relatively quickly as they start to work with more complex mathematics. For example, if a student can’t produce the product of 6 x7, then they will certainly struggle with finding equivalent fractions with denominators of 6 and 7. However, just having these facts memorized is NOT going to ensure future success, in fact, some recent research has demonstrated that students who are great memorizers often struggle the most with upper level mathematics (Boaler, 2014).
In addition, students with basic number sense will be able to solve 6×7 and will be well equipped for upper levels of math. The National Council of Teachers of Mathematics states: “Students with number sense naturally decompose numbers, use particular numbers as referents, solve problems using the relationships among operations and knowledge about the base-ten system, estimate a reasonable result for a problem, and have a disposition to make sense of numbers, problems, and results.” (NCTM, 2014). These are the skills that will set our students up for success.
Using manipulatives, especially the base 10 blocks, helps students to develop strong number sense and helps them to create strategies that will increase their fluency with the basic facts. Using these blocks to explore operations with whole numbers in elementary school benefits students by: reinforcing place value, increasing number sense, developing strategies for fluent computations, understanding the connections and relationships within the place value system and within operations, and it provides them with a useful problem solving tool. For example, use the base 10 blocks to explore the relationship between single digit number multiplication and multiplying multiples of ten.
If 3 x 4 = 12 because 3 groups of 4 is equal to 12, then 3 x 40 is 3 groups of 40 and thus 120 and 3 x 400 is 3 groups of 400 which is 1200. Using the blocks students can discuss the similarities between these 3 sets of products; each one has 12 blocks. In the first example, there are 12 one’s, in the next example there are 12 ten’s and in the final example, there are 12 hundred’s. Students can then realize that these operations are very similar but instead of teaching them to just multiply the number and add a zero or two, students can discover for themselves that the products will indeed be similar to 12 but ten times bigger or a hundred times bigger, depending on the sizes of the blocks. This is developing number sense. Telling them to add a zero does not.
Greater Understanding of Math Concepts
Base 10 blocks are also extremely helpful in helping students understand decimal numbers and operations. We often teach decimal numbers using a place value chart (which is symbolic) and students don’t ever get to see how small a thousandth is compared to a tenth or hundredth. The blocks help students to understand that the relationships in the place value system are consistent in decimal and whole numbers and they understand why a tenth is called a tenth as well as why there is no ‘oneth’. The blocks allow students to discover for themselves why operations with decimals are so similar to multiplying whole numbers and this way they are building number sense and they stop seeing decimals as a completely different topic than whole numbers but rather see how they are connected. The more connections students can make to prior learning, the stronger the new learning becomes.
I could go on about the virtues of cuisenaire rods, fraction circles, pattern blocks, algebra tiles, geoboards, etc. and I will (look for this in future posts) but suffice to say, the principles are the same for any of these manipulatives; students are learning far more than just computational skills, they are learning mathematical skills. Manipulatives are intended to be used to introduce a concept and then once students have a firm conceptual understanding, we start to move into the pictorial representation and finally into the symbolic. It is important to be aware that this progression often isn’t linear and is rarely the same for all students. For example, some students may need to use the manipulatives a lot longer than others. Some may not even be able to transition to the symbolic in the year you are teaching them. I would much prefer a student with some conceptual understanding who can demonstrate this with manipulatives than a student who has memorized an algorithm but has no idea what it actually means, because they are not going to be able to apply it or build upon it. By creating a culture in your classroom that celebrates the variety of learning styles and speeds, students will be perfectly comfortable using manipulatives even if the rest of their classmates have moved into the symbolic.
There is a reason why using manipulatives is in our curriculum; both brain and educational researchers agree that in order for students to really learn math effectively (not just computational skills but all aspects of math), we must provide them with adequate tools so that they can develop conceptual understanding. We do this by allowing them to discover for themselves the vast relationships and connections. When students can make these discoveries and explain their understanding, they are far more likely to retain their learning. They are also far more engaged and see mathematics as more relevant and more creative.
Take Aways and Tips
1.) Math classes need a make-over and using manipulatives is one way to begin. Some teachers believe they are not worth the time it takes to use them and that students are too distracted by them to learn. In the past 7 years, I have not found this to be true at all. They are well worth the time it takes; what’s the point in spending hours practicing a skill that they don’t understand and will forget as soon as the test is over? Time is better spent developing understanding and meaning, otherwise the learning will likely never make it to the long term memory. As far as classroom management, here are some tips:
2.) Allow students play time with the blocks before using them for math. A minute or two will suffice for middle school students. I know some teachers that allow more play time with younger kids as there is a lot to be learned through play. I encourage parents to buy manipulatives so kids can play with them at home.
3.) Have paper versions of the blocks for those students who cannot use them properly. I have only had to take away the blocks a small handful of times before they learn to use them for math rather than playing.
4.) The rule is: no hands on the manipulatives while anyone is sharing out or when the teacher is talking (they can cross their arms, sit on their hands, push the blocks out of the way etc.).
5) Be patient. It will be noisy and they will fiddle but it is worth it. The learning is worth it. We have this constant pressure to get through the curriculum but think about how easy it would be get through your grade’s curriculum if you didn’t have to reteach all the previous grade’s learning. If students develop strong number sense and conceptual understanding, they will retain more of their learning, thus allowing you to move faster through related topics. It pays to slow down at the beginning.
6.) Have fun with it. Embrace the creativity and the new way of thinking about math. Allow students to share their ideas and thinking as they pick this up faster than most of us do (they don’t have decades of unlearning to do before learning this way).
7.) Make sure you are well prepared. It is an awful feeling when you try to wing it and then it bombs and it leaves you and the students feeling confused and frustrated. Sometimes this happens but use it as a learning experience; full understanding will likely not happen in one lesson.