Using Manipulatives for Pattern Recognition in Mathematics

Helping students understand the relationship between patterns and algebra can be an abstract progression. Using manipulative to help students visualize the patterns can be an incredibly useful tool. If teachers can start getting students to visualize the progressions of certain patterns and develop their skills in visualization the transition into algebra will be much smoother.

The following pattern is an example of this type of relationship:

Photo by author.

When I look at this pattern it is easy to develop an algebraic expression to determine the relationship. The expression n+4, where n=term number represents this.  Although it is easy to develop this expression it is important to take the time to look at this relationship and see why this expression makes sense. If we can have students focus on the fundamentals of developing this expression. If we can have students look at the diagrams and determine which parts are growing and which parts are consistent they can easily see relationship. Recreating these patterns with manipulative gives students the opportunity to visualize the progressions. The green triangles remain constant at 4. The purple square increases by one for each term; starting at one. 

The exciting part of this is that with repetition and practice students will be able to identify more complex problems and how they relate to linear relationships. 

Photo by author.

After practicing pattern recognition and developing algebraic expressions students can begin to relate these activities to graphing linear relationships. Take the pattern to the right for example. While recreating the patterns students will be able to develop the algebraic expression to be 3n+4. The difficult part in this example would be seeing that the centre square is also part of the fixed triangles. It would be through the practice that students would become accustom solving this. 

Photo by author.

From here, students can use manipulatives from the patterns on graph paper to visualize the growth.  Next they can track the progressions and relate the this growth back to the algebraic equation. This lays the frame work for discussions surrounding graphs as well. The relation between term 0 and the y-intercept is is easily seen. Extrapolation is also evident while using this method. 

Why Are WE Scared To Estimate?

My instructor brought up a great point the other evening about students struggling when asked to estimate answers to basic mathematic questions. (L. Surrturmm, personal communication, September 14, 2015) This really got me thinking and the more I thought about it the more I could relate to the issue at hand. Through this post I hope to shed some light on the reasoning and a possible strategy to correct this.

retrieved from: http://images.clipartpanda.com/estimate-clipart-math_estimate.gif 

I want you to think back to an experience I'm sure most of you all had in a math class. We've afll been given a math question asking us to first estimate the answer than compute it after to see how close we were. For example:

Question:

"If Jenny gets  $.47 for every Apple that she sells how many dollars while Jenny have if she sells 30 apples per day and works 90 days in the summer?"

Answer:
Estimation:___________
Computation:_________
Show your work:

_______________________________________________________________________

Now if you were like me I'd be more concerned about making sure my estimation was as close as possible to the correct answer because I was worries that the teacher would think I didn't have a good understanding of the content if it was far off. For that reason my approach to this question would be to fill out the show your work section first, then the answer. Finally I would put a number not far off of the correct answer as my estimate.

Looking back on this and trying to understand why I'd (I wanted to say children but that wouldn't be accurate because I'm not sure I'd be any less of a culprit to than I was back then) approached this question this way I came up with a couple reasons:

• I was caught up in the idea that I always had to have the correct answer.
• I was worried the teachers would be disappointed in me if the estimate was far off.
• I didn't understand the importance of developing strategies related to estimation.

Now that I've given this a bit more thought with an education, development lens I'd like to explain an option of how I'd have wished I'd approached this question:

First I'd highlight the important parts:

"If Jenny gets  $.47 for every apple that she sells how many dollars while Jenny have if she sells 30 apples per day and works 90 days in the summer?"

Next I would round the numbers to something more manageable:

-90 days in the summer would become 100 days.

-30 apples per day could remain 30 apples/day.

-$.47 for every apple would become $.50/apple.

From here I would say 30 apples/day for 100 days is 3000 apples. Then I'd think if Jenny makes 1/2 of a dollar per apple she would have half as many dollars as she sold apples. That would be 3000/2 which is an easy 1500. 

Making my estimation become; $1500 for Jenny over the summer.

________________________________________________________________________

If we look at this suggested process for estimation it resulted in an answer that was only a few hundred off and a 15.4% error - which if we consider the fact it took hardly any mental math to solve is pretty impressive. 

If we take the time to encourage students to think of ways to answer simple questions, like this apple problem, through basic reasoning and utilizing estimation we will be developing their adaptive learning skills.  In my opinion this is one of the best ways to set the foundation of have students thinking outside the box and develop creative solutions to complex problems. 

Our first steps to move students in this direction is slowing down our lessons and providing basic techniques creating manageable problems. We want to give the students the opportunity to make educated guesses. Once students start developing some skills and strategies for estimation then introduce more complex, challenging problems. 

________________________________________________________________________

Feel free to comment and let me know you opinions on estimation.

• Did you have any experiences similar to the one I outlined?
• Did you have an experience very different from this one?

Intermediate/Senior Physical Education and Mathematic Education

The posts I have been uploading thus far have been based around education in general.  As an intermediate/senior teacher candidate with teachables within Mathematics and Physical education I figured it would be beneficial to introduce myself in regards to these.

First of all; my name is Mr. Studenny and I am a teacher candidate at Brock University's Faculty of Education.  My first teachable is in Physical Education and my second teachable is in Mathematics. As a teacher candidate I plan to bring a combination of the skills I have learnt through these domains to the classroom.  An area that has really sparked my interest through my undergraduate degree has been in creating an engaging inclusive environment for my students. I find that the creation of a sense of community in the classroom can totally change and shape the learning environment in a very successful way.

I am particularly interested in how students react positively to a classroom that has a strong sense of community. I feel this helps engage and motivate students. Furthermore, it also sparks initiative and creativity in them. I view each an every one of these traits as a fundamental important characteristic of an engaging, successful classroom.

I can not wait to apply these principles to the physical education and mathematics curriculum.  I completely understand the struggles these will bring during the next few years of my professional life. I'm excited to face these challenges and see where they take me.

Cheers,

Mr. Studenny