A teacher stood before a class of thirty senior mathematics students. Before he passed out the final exam he stated, “I have been privileged to be your instructor this semester, and I know how hard you have worked to prepare for this test. Because I am confident that you know this material, I am prepared to offer an automatic B to anyone who opts to skip taking the final exam.”
There was great relief for some. A number of students went right for the offer and thanked the teacher.
“Last chance,” said the teacher and one more student went.
The instructor then handed out the final exam, which consisted of two sentences. “Congratulations,” it read, “you have just received an A in this class. Keep believing in yourself."
This problem was given to 1st and 2nd graders and many tried to answer the problem by using the numbers in the question and picking a mathematical operation. If math is too focused on memorization then students are not seeing the power of mathematics. The following points should be emphasized with students when learning math.
· Everyone can do math!
· Emphasize reasoning over memorization
· Encourage multiple strategies and ways of thinking
· Math is not about how quick problems can be done but shortcuts with understanding are great!
· Encourage discussion and exploration
· Math should foster curiosity
· Math should be relevant and realistic
The following video has a similar problem done with 8th graders
The following video has more details on the, how old is the captain problem.
In this game sixteen linear graphs are given. One student selects one of the graphs and the other student asks yes or no questions to determine which graph has been selected. Between games students are shown questions that other students ask. The teacher also is able to view and have a record of all questions asked in each game.
Table 1 has the initial questions that were asked by 4 of the groups. I analyzed the data with an interpretative approach by looking at the ways in which students used mathematical vocabulary in the game.
After playing the game several times, the students discussed what quality questions to ask and
strategies for asking the least amount of questions. Several questions appeared in common in the
groups: “Is your slope positive?” “Is the slope negative?” “Is your line horizontal?” “Is your line
vertical?” “Does your line go through the origin?” Groups also came up with questions of what
quadrants the line crossed through, though not all groups used the term “quadrants.” Through playing
the game and subsequent discussions, students were able to make use of mathematical vocabulary
including slope, positive slope, negative slope, horizontal line, vertical lines, origin, and quadrants.
Stohlmann, M. (2020). Integrated STEM education through game-based learning. In A.I. Sacristán, J.C. Cortés-Zavala, & P.M. Ruiz-Arias (Eds.). Mathematics Education Across Cultures: Proceedings of the 42nd Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA). (pp. 2238-2242). Mazatlán, Mexico: PME-NA.
What happens when you want something that someone else wants as well? This excellent book reinforces the ideas of sharing and compromises. Children will be intrigued by the story and the questions that are integrated with the pictures. Children will remember that sharing is caring!
A useful sequence of experiences when working with measurement and volume is a three-step experience described in the table below.
For step 1, students can do a comparison activity in which they determine which of two glasses holds more juice. Through this book children can develop informal notions of volume and also that there are different ways of sharing items. The pictures and questions in the book provide opportunities for rich discussion related to sharing and mathematical ideas!
The game, Bigger or Better, is now a book! Each team starts with the same item. They then must go out and find people that are willing to trade something for the item. The goal is to end up with the biggest or best item. Which team will win? This fun and lively book will keep children engaged! Check the back of the book for ideas on how to develop children's number sense for fractions!
When two fractions are not equivalent but are parts of the same whole or unit, there are several ways to find which is greater through comparison. It is important that the fractions refer to the same whole or unit though. This could be comparing a fractional amount of a pizza to a fractional amount of a pizza the same size. Other examples that are possible are to compare length in feet versus length in feet, area in feet squared versus area in feet squared, or comparing weight measured in pounds. However, comparing fractional amounts of two different units can be difficult to do. For example, 1/3 of a cake versus 4/5 of a chocolate bar. 4/5 is a larger number than 1/3 but the cake may actually be bigger in size compared to the remaining chocolate bar.
Comparing fractions using concepts
A recent review of studies that involved game-based learning found that most of the games used in the studies involved drill and practice. In drill and practice type games, students only receive feedback on if answers are correct or incorrect and do not receive support for conceptual understanding. These types of games also emphasize that mathematics is about speed and focus on memorization of ideas instead of conceptual understanding. Game based learning for mathematics should move beyond drill and practice.
My principles for technology game-based learning ensure that games are selected and implemented with best practices for teaching mathematics in mind. First, the technology integration should allow for significant task redesign or the creation of new tasks that would not be possible without the technology. Second, the tasks used should be worthwhile tasks. These tasks have no prescribed rules or methods and there is no perception that there is a specific “correct” solution method. Third, the tasks should be aligned with grade-level standards. Fourth, the tasks should enable students to work with multiple representations. Fifth, the technology should provide students feedback. Finally, the tasks should be open-ended and allow for discussion and multiple solutions (Stohlmann, 2019). When structured well, technology-based mathematics games can engage students in mathematics and help develop their conceptual understanding.
Stohlmann, M. (2019). Integrated steM education through open-ended game based learning. Journal of Mathematics Education, 12(1), 16-30.
Students should do mathematical work that is challenging with feedback and any needed scaffolding. Students should have the opportunity to demonstrate understanding in different ways. High expectations are key.
Students should see the power of mathematics and understand how mathematical knowledge is relevant to their current and future lives.
When students know that their teacher believes in them, cares about them, and wants them to succeed it makes all the difference.
In context game-based learning games are used as an interesting context to pose mathematical problems. When students play the mathematical context games, they are not doing mathematics but do solve mathematical problems related to the games. For example, in the water bottle flipping activity students play a game to see how many times in a minute they can flip a water bottle and get it to land straight up.
In the activity, students do five one minute trials in which they record how many lands of the water bottle they can make in one minute. The world record for this is 47 lands in one minute. Based on the five trials students then calculate their average number of lands per minute. Students then fill in a table based on this average and answer follow-up questions (Figure 1). The activity has students work with proportional and linear equations through tables and equations. In the whole class discussion after students have answered the questions, connections can also be made to the graphs of students’ equations and interpreting the graphs in the context of the game. Questions can also be posed to compare equations in regards to slope.
The context of water bottle flipping engages students and allows for interesting questions to be asked. Students are able to interpret mathematical answers in the context of the game and make connections between representations. Another example of context game-based learning is the paper basketball activity. In this activity students estimate and calculate how many paper balls will fit into a bucket. After doing the mathematical work students then race to see who can make the most shots of paper balls into the bucket in a minute. This game is engaging and motivates the mathematical work through different representations that incorporates measurement, mean, volume of spheres, and linear and proportional equations. Doing mathematics in the context of games engages students through interesting mathematical work, movement, and healthy competition.
Class closers allow for a quick assessment to see what students have learned and the teacher can connect the ideas back to the stated objectives from the start of the class to review if the objectives were met. Below are some basic examples and following that are some more creative ideas to mix things up every now and then.
Of all the fourth graders at Fairview Elementary the twins, Tony and Tina, were known for their ability to make friends. They enjoyed school but struggled with mathematics. After learning more about the importance of mathematics, Tony and Tina strive to help their classmates have the right mindset. Find out how Tony and Tina's ideas lead to mathematical success for themselves and their classmates!
“You teach math? Oh I never got math.” I have heard the following comment so often, that I decided I would try to do as much as possible to change this sentiment. Mathematics is a subject that has the stigma that some people are “just not math people”. This is a dangerous idea that needs to change. While everyone will not go into a math heavy career, everyone is capable of doing mathematics. The life skills including teamwork, communication, and being synthesizers of information that students can develop from mathematics will help in any career as well.
Mathematics education can be improved with little things that could cause a tipping point. A tipping point is the moment at which ideas and messages can spread very quickly to cause change. Eliminating the phrase, “I never got mathematics” and replacing it with the phrase, “You can do math. I can do math” is a good start to improving mathematics education for all. Beliefs are the best indicators of the decisions that people make over their life-time. By giving children the right mindset at an early age, they will be set up for success!
Christian, author, and professor of mathematics education.