Wednesday, September 28, 2011

Constructivism in Practice

As a math teacher, it can sometimes be difficult to think of ways for students to generate and test their hypothesis.  In the book entitled Using Technology with Classroom Instruction that Works, one of the recommendations mentioned was problem solving (Pitler, Hubbell, Kuhn, Malenoski, 2007).  In math, my department consistently incorporates problem solving in every lesson.  This can be in the form of solving algebraic skill problems or solving real world word problems. 

My students are currently learning the rules of exponent as well as how to simplify problems containing exponents.  With the knowledge they gain while learning basic exponent rules, students may have a better understanding of the skill once we begin lessons on square roots.  For instance, students will devise a list of integers that are raised to the power of two (or squared).  Then, students will work backwards to devise a list of square roots.  Students will generate and test a hypothesis as to what the actual outcome would be by estimating each problem.  They will test their hypothesis by actually working the problem to see how close the actual answer was to their hypothesized answer. 

In order to enhance students’ learning, technology could easily be utilized.  According to Dr. Orey (Laureate, 2011), students should be engaged in learning and the process of creating an artifact to share with others to follow the theory of constructionism (Laureate Education, Inc., 2011).  For this particular lesson involving exponents and square roots, students could create a spreadsheet via Microsoft Excel.  Students could design a table with the problem in one column, their hypothesis in the next column, the actual answer in another column, and the last column would involve the difference from their hypothesized answer and the actual answer.  Then, students could create a line graph that represents the difference from their hypothesis and the actual answer.  This way, students can visually see how off or how close they were to generating the correct answer for each problem.  After generating their graph, students can share their findings with their collaborative groups.  Students can compare their differences and distinguish which problem(s) the majority of students were missing.  Finally, collaborative groups can work together in order to formulate a plan for determining the correct solution to those problems which had the most errors. 


Laureate Education, Inc. (Producer). (2011). Program five: Cognitive learning theory [Video webcast]. Bridging learning theory, instruction and technology. Retrieved from

Pitler, H., Hubbell, E., Kuhn, M., & Malenoski, K. (2007). Using technology with classroom         instruction that works. Alexandria, VA: ASCD.


  1. I too had a similar use for Excel. I like the way a user can easily graph their analysis. It provides a much needed alternate venue for others who get lost in numbers an effective method to visually interpret data.

    I am wondering, however, if you have thought a little bigger scale with respect to real-world problems. I teach high school math and try to incorporate word problems every day as well. For me, it becomes a little mundane. I think that the our resources may have implied a grander scale of problem-solving. For instance, have you thought of collaborating with your biology/chemistry counterparts to assist in tracking the exponential growth of bacteria? How about the reaction rates between two chemicals? I suppose my point is that cross-curricular collaboration can provide real-world applications to the stale classroom we often teach within. I admit, I haven't tried these ideas yet. I am still developing some of the lesson aspects but I have it in the "hopper" ready for the right opportunity. Currently, I use other project of that nature that synthesizes several sections and uses outside resources. I think when we can provide students opportuniites like this, they can truly learn how to use the content we are trying to teach them and that it actually has a place in everday life.

  2. Malissa,

    I really enjoyed reading about how you would incorporate technology into a math lesson about exponents. I agree with Dr. Orey on the fact that students should be engaged in their learning. I find that when students are engaged they have a better chance of remembering the information. I really like to incorporate as much technology into my lessons as possible because the students enjoy it so. I just wish that I was more familiar with Microsoft Excel as you seem to be. Great post and a great idea!


  3. Malissa-
    I agree, sometimes using technology with math lessons seems more difficult than other subject areas. One way I am looking to incorporate more of it in my third grade math class is once students understand the concept of multiplication and division, I will give them real-life word problems that require multiple steps. Then they have to guess what they think the answer will be before solving using their estimation and rounding skills. Then as they solve the problem they must be able to explain how they solve it in written form. This will become a class blog where students explain how they solve the problem. This incorporates not only technology but also critical thinking skills with writing skills as well.

    Another aspect you may consider for your class is in geometry. Given a shape, students must identify the shape. Then test their hypothesis with the rules that define that shape and give support for their answer. You could also give them a shape and give it an incorrect name and they have to give evidence why it is not that shape. These are a few ideas I have come up with to try with my students this year. Good luck!

  4. Joe,

    I really like your cross-curricular ideas for generating real world problems in my math class! This may also capture the attention of students who are not particularly fans of algebra, but may really like social studies, or science (even though they will ultimately be using algebra to solve these problems). We typically keep the algebra in algebra class and the biology in biology class. Your idea of presenting word problems in this manner is an exceptional suggestion that I will definitely take into consideration when implementing constructivist learning strategies into the lessons. Thanks, Joe!

    ~ Malissa

  5. Heather,

    Thank you for your comment! While I am no expert at Microsoft Excel, I am still learning as I go. My school began using began using Excel for student standardized test data collection. I just learned how to use the "filter" function. I am realizing that there is more to this spreadsheet software than just generating tables and graphs. You will get there, just keep practicing!

    You're right - students do seem to be more engaged when they are using technology. Who knows, maybe they can share what they know with you! That's the cool thing about technology and everyone's ability levels. Students are excited to share what they know!

    ~ Malissa

  6. Judy,

    What a fantastic way to incorporate writing into your math class, not to mention the critical thinking skills that students will be able to reflect upon on while discovering the solution to the real world problem. I really like how you will incorporate a blog into your lesson as well.

    I really like your suggestion for correct names of shapes. Often in my geometry unit, students confuse typical parallelograms based on their unique properties: squares, rectangles, rhombuses, kites, and trapezoids. I will definitely take your idea into consideration when I devise this constructivist lesson. Thanks, Judy!

    ~ Malissa