Differentiating Instruction: Open-Ended Questions

This week in class, we spent a long period of time discussing differentiating instruction through the use of open-ended questions. In small groups, we were asked to solve the following question,

“create three problems that seem really different that could be solved by calculating 2012-1987.”

http://writedge.com/benefits-asking-open-ended-question/

One of the solutions my group came up with included, “Sally was born in 1987 and Joe was born in 2012. How much older is Joe?” Other groups had answers involving money, number of children in schools, wedding anniversaries, and so on. Each group had their own unique way of approaching the question, all eventually providing correct responses. Similar to this question, open-ended questions have no fixed answer, can be solved using different strategies, are accessible to different ability levels and empower students to make their own mathematical decisions and reasoning. As Amy Lin mentioned in class, open-ended questions have the potential to be used at any point during a lesson.

Reflecting upon my mathematical experiences in both high school and university, I was presented with little opportunity to work with open-ended questions (as I am sure many others can relate to this). In high school, rarely in the ‘communication’ section on a test would our class have the opportunity to determine numbers to work with. In University math classes, we were always presented with structured questions that had predetermined answers, or in other words, closed questions. Closed questions are usually content specific and do not foster student creativity, innovation or reflective dialogue. Although I was a successful student, I always found myself hesitant to answer the teacher because I was aware that he or she had an exact process and answer they were looking for.

http://firstgradebloomabilities.blogspot.ca/2015/01/the
-value-of-open-ended-math-questions.html

So why do open-ended questions work? How do open-ended questions foster this deeper-level of thinking we are constantly hearing about? I really enjoyed how an article by Responsive Classroom explains why open-ended questions are so powerful. The article mentions that such questions take learners through a cycle of “wonder, exploration, discovery, reflection, and more wonder, leading them to increasingly complex knowledge and sophisticated thinking.” When teacher’s use open-ended questions in their classroom, it provides students with a sense of autonomy, belonging and competence because they believe that the teacher trusts they have good ideas. As demonstrated in our class this past week, open-ended questions also foster communication and collaboration skills. After discussing the activity with the rest of the class, students can also take away multiple strategies and solutions to the question.  

https://www.tes.com/lessons/lrDlRlZiTH8JWg/differentiated-instruction

Over the past few weeks I have grown to appreciate the process of differentiating instruction in a mathematics classroom. I intend to use open-ended questions in my own teaching practices to help foster student engagement and success, but more importantly to help meet the needs of various learners. In class we also briefly discussed alternative ways to assess students as opposed to traditional paper and pencil techniques. Some methods include: conversations, project based learning or applied projects, group discussions, and interviews. 

I am excited to challenge myself to further extend my knowledge on differentiating instruction in mathematics. I challenge you to do the same!

Thanks for reading,
Rachelle

Openended questions (2007). Retrieved from https://www.responsiveclassroom.org/open-ended-questions/

Manipulating Math

I hear and I forgot

            I see and I remember

                        I do and I understand

                                            -       Confucius

This week in class, we explored the topic of implementing manipulatives in the classroom. As a group, we (the teacher candidates) had the opportunity to solve various mathematical problems using objects such as candy, ropes and algebra tiles. Although we are a very competent and intelligent group of future math educators, it was evident that the use of manipulatives fostered engagement and success. Manipulatives are not just useful for elementary students, students whom are struggling or those in the lower streams of math; manipulatives are for everyone!

So why manipulatives?

Manipulatives are concrete objects used as a teaching tool that allow students to explore a concept in an active, hands-on approach. Traditionally, manipulatives are used in the classroom to introduce, practice or reinforce a concept. Common classroom manipulatives include: algebra tiles, base ten blocks, connecting cubes, pattern blocks and so many more. The use of manipulatives allows students to reach a deeper level of understanding because they are constructing their own cognitive models for abstract ideas and processes. More importantly, manipulatives engage students both visually and physically.

An article from Hand2mind discusses how manipulatives move students through the three stages of learning (concrete to representational to abstract). Students begin using concrete materials to solve problems by looking for patterns or generalizations. In the representational stage, students record their work by sketching representations of their manipulative models. Lastly, students move to the abstract stage where the objects and pictures are connected to the abstract numbers and signs of arithmetic. While moving through these stages, students gain an understanding of the mathematical problems through their direct experiences in the concrete and representational stages.

Movingwithmath.com

What are the benefits?

Which sounds more interesting: passively copying down the problems your teacher is writing on the board, or collaborating with your classmates using three-dimensional shapes to determine the surface area and volume? Manipulatives can be used both individually or amongst a group of students. When implemented through a collaborative approach, students engage in conversations where they are able to verbalize their mathematical reasoning using a common language. As students work through the problems, they gain self-confidence and pride through the process of solving the problem using a method that “works” for them. As I mentioned in my last post, I believe building confidence amongst students is one of the most important steps in achieving student success. I came across this video on using manipulatives for multiplication and thought it was a great demonstration of a student strengthening their understanding. 

https://www.youtube.com/watch?v=UKlz2aWa_Mg

From personal experience, the constant use of paper and pencil assignments in math class can become very tedious and boring. Manipulatives are fun and for some students they represent “play” as opposed to work. In class, we had a worksheet that involved counting candies and chocolates with their associated cost. Without realizing it, we were in fact solving a system of linear equations. As a future educator, I intend to use an ample amount of manipulatives that apply to all students. We live in a very technological based society, and thus manipulatives could also include technology (another aspect of my teaching I hope to further develop).

Ispeakmath.org

Personally, I wish I had the opportunity throughout my educational experience to use more manipulatives. As a student in the academic stream, many of my teachers possessed the preconceived notion that our class did not require manipulatives to aid our learning. This past year, I had first hand-expereince with the result of using manipulatives. I was tutoring a younger student in algebra and we were working on collecting and multiplying like terms. Without the use of manipulatives, the student had great difficulty grasping the concept. With the use of the algebra tiles, her engagement and motivation immediately shifted. It was clear she was a visual learner and the use of algebra tiles significantly increased her understanding. I can only assume that this outcome would be very similar for many other students. Algebra tiles can also be used for factoring, completing the square, etc. Edugain explains the use of many different manipulatives and provides activities and tip sheets for each type. 

Of course, it is important to note that educators should not over-use manipulatives in their classroom. As this course continues, I am looking forward to further developing my knowledge on the use of manipulatives. I am excited to have this as one of the tools in my teaching toolbox. 

That’s all for now,
Rachelle

Why Teach Mathematics with Manipulatives? (2016) Retrieved from http://www.hand2mind.com/resources/why-teach-math-with-manipulatives