-What was the big idea we worked on today?

-What did I learn today?

-What good ideas did I have today?

-What did I struggle with today?

-Where could I use the knowledge I learned today?

-What questions do I have about today's work?

-What new ideas do I have that this lesson made me think about?

This is a good open-ended 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. 

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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?

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 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.

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!

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.

Reference:
Stohlmann, M. (2019). Integrated steM education through open-ended game based learning. Journal of Mathematics Education, 12(1), 16-30.