This is a good openended problem that creates quality mathematical discussion. The median and the mean of the scores for Jenna and Kim are the same, which can require students to think of other possible solutions. A benefit of this task is that there is no one specific "correct" solution. There is also no prescribed method for solving the problem. Incorporating these types of problems leads to increased engagement and discussion in class.
 There were five judges for a figure skating competition: The top two competitors were Jenna and Kim. They received the scores below. Jenna 8 6 10 9 7 Kim 9 9 7 8 7 Who should win?
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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. Reference: Stohlmann, M. (2020). Integrated STEM education through gamebased learning. In A.I. Sacristán, J.C. CortésZavala, & P.M. RuizArias (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 (PMENA). (pp. 22382242). Mazatlán, Mexico: PMENA. 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! Volume A useful sequence of experiences when working with measurement and volume is a threestep 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! Comparing 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 gamebased 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 gamebased 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 gradelevel standards. Fourth, the tasks should enable students to work with multiple representations. Fifth, the technology should provide students feedback. Finally, the tasks should be openended and allow for discussion and multiple solutions (Stohlmann, 2019). When structured well, technologybased mathematics games can engage students in mathematics and help develop their conceptual understanding. Reference:
Stohlmann, M. (2019). Integrated steM education through openended game based learning. Journal of Mathematics Education, 12(1), 1630. Rigor
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. Relevance Students should see the power of mathematics and understand how mathematical knowledge is relevant to their current and future lives. Relationships When students know that their teacher believes in them, cares about them, and wants them to succeed it makes all the difference. In context gamebased 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 followup 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 gamebased 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.

Micah StohlmannChristian, author, and professor of mathematics education. Archives
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