In June of 2016, I took two graduate level mathematics classes (Number Theory and Discrete) at Bemidji State University (BSU). BSU’s focus is on improving mathematics instruction in the K-12 classrooms vs learning high level mathematics that would never apply to teachers as they work with K-12 students. I learned some really good ways to help students understand some mathematical concepts either intuitively or with manipulatives. Some of the topics that stuck out to me are:
- adding, subtracting, and multiplying integers
- dividing rational numbers
- greatest common factor
- least common multiple
- permutations and combinations
- proofs of divisibility rules (you may think this topic would be boarding but it was quite interesting)
The two classes I took were back to back with class officially starting at 8:00 and going to 1:30 with a 30 minute lunch (provided) but most people were their 15 to 30 minutes early. You may think that 5 hours of class is a lot and it would be if it were just straight lecture but BSU’s approach of asking questions, giving you time to think individually, discuss in small groups, share your ideas at the board, with only a little lecture here and there made class go pretty quickly. The approach that BSU used in their classes had some characteristics that reminded me of Number/Math Talks. Many of topics they shared with us to help improve the instruction of mathematics at the K-12 level arise from either Nation Science Foundation curriculums or their decades of teaching experience.
Below is a brief outline of the approach or approaches taken and a link to a video to help you understand it better. Some of the topics apply only to upper elementary and middle school while other topics have aspects that can be applied in algebra and other high school mathematics classrooms.
Adding and Subtracting Integers
The approach that BSU shared came from IMP (Interactive Mathematics Program). The idea of adding and subtracting integers is rooted in the story of “The Chef’s Hot and Cold Cubes”. Part of the story is shown below.
In a far-off place, there was once a team of amazing chefs who cooked up the most marvelous food ever imagined.
They prepared their meals over a huge cauldron, and their work was very delicate and complex. During the cooking process, they frequently had to change the temperature of the cauldron in order to bring out the flavors and cook the food to perfection.
They adjusted the temperature of the cooking either by adding special hot cubes or cold cubes to the cauldron or by removing some of the hot or cold cubes that were already in the cauldron.
The cold cubes were similar to ice cubes except that they didn’t melt, and the hot cubes were similar to charcoal briquettes, except they didn’t lose their heat.
If the number of cold cubes in the cauldron was the same as the number of hot cubes, the temperature of the cauldron was 0 degrees on their temperature scale.
For each hot cube that was put in the cauldron, the temperature went up one degree; for each hot cube removed, the temperature went down one degree. Similarly, each cold cube put in lowered the temperature one degree and each cold cube removed raised it one degree.
This story sets the stage for helps students add and subtract intuitively and with manipulatives that represent a hot cube and a cold cube (the “cubes” could easily be red and blue counters or could be represented as just an “H” and “C” but giving the students something physically to touch and move is preferred). Here are three links to some videos I made to help others use this approach: Hot Cubes and Cold Cubes, Hot and Cold Cubes Practice Problems, and Summarizing the adding and subtracting integers
Do not start out by showing the students how to model all eight types of problems but only show them how to model one or maybe two problem then let them try to figure out how to model the other problems. It is good to let students be perplexed but be watchful so you can prevent the perplexity from switching to frustration. After some time of the students working on one problem ask some groups that were able to model the problem to go to the board and demonstrate it to the class to help other groups model it. One major mistake that most of us teachers do is to go to quickly to the summarizing of the adding and subtracting rules or letting one of the top students share the patterns (rules) they noticed for adding and subtracting integers too early. It may take a week and a half of having the students playing with the hot and cold cubes modeling the adding and subtracting of integers before you start to summarize the rules/patterns for adding and subtracting integers.
The multiplying of integers uses the idea of hot and cold cubes or could be done with two colored counters. The approach used in the video link below starts with multiplication being repeated addition. Start by modeling one problem and letting the students see if they can model the rest. Here is a video I made on this concept Hot Cubes and Cold Cubes Multiplication .
Dividing Rational Numbers
I have only seen one way in which students can use manipulatives to model dividing of rational numbers that works wells and that is the approach that I am sharing with you. This approach uses unifix cubes to model the dividing of rational numbers. Most people like a story and there is a short story that leads the students into modeling the problems. Start with an easy problem then continue to asking more and more complex questions. Here is a video I made on this concept Dividing Rational Numbers with Bricks .
Some of the above topics do not apply a lot to high school students but the next few topics do have some application algebra and other high school.
Greatest Common Factor and Least Common Multiple
The greatest common factor and least common multiple videos have approaches that will help students in algebra. Start by sharing why you would want learning these concepts (LCM is used for adding and subtracting fractions while GCF is used for simplifying fractions). With the approaches shared with us on GCF and LCM both start with using the words of Greatest Common Factor and Least Common Multiple. For GCF, list all the factors then identify all the common factors followed by finding the greatest common factor and the LCM does a similar approach. But what I really like is the second approach of each that uses the prime factorization approach. Most math teachers in middle school and high school will write the prime factorization of 48 as 24*3 but that approach leads to many misunderstandings or questions for students like “Why for Greatest Common Factor do we use the smallest exponent when we have a common factor show up for both numbers?” or “Why for Least Common Multiply do we use the largest exponent when we have a common factor show up for both numbers?” But the second approach of prime factorization of writing out 48 as 2*2*2*2*3 is so intuitive to students. Here is a couple of videos I made on these concepts Greatest Common Factor and Least Common Multiple .
Permutations and Combinations
This approach starts out asking students how many ways you can arrange a simple word followed by slightly harder words then gets into words with repeated letters which the way it is done is fairly intuitive. The permutations of words with repeated letters leads naturally into combinations. The second video extends the idea of combinations by asking a question of “How many ways can parents have three girls in a family of five?” Most of us who teach math know this is a basic combination problem but most students will think of it this way if you ask them this question the day after you introduced combinations. This basic combination idea can be extended even more and tied into Pascal’s Triangle. Here is a couple of videos I made on this concept Permutations and Combinations and Ways of having kids in a family and Pascal's Triangle .
To really challenge some of your top students, ask them “How many ways can you form groups of four out of a class of 24 students?” If you think the answer is a simple combination of twenty four choose four then you are thinking incorrectly like I was initially. You may want to let the students struggle with it for a while then emphasize the “s” in groups, “How many ways can you form groups of four out of a class of 24 students?” If needed follow that clarification with, “How many ways can you form one group of four out of a class of twenty four students?”
Proofs of Divisibility Rules
I would not do these proofs with elementary or middle school students since they are algebraically based but I think they would be good for high school students. High school students have been using divisibility rules for years but they know have the algebra background to be able to prove the divisibility rules. I would start by modeling one or two proves to my students then I would let them try one that is a little harder as they work in groups. They will likely pick up on some patterns for doing these proofs. These proofs are a good extension for Algebra 2 or Pre-Calculus students. Here is a couple of videos I made on this concept Proofs of Divisibility Rules 2, 3, 4, 5, 7, 9 and Proofs of Divisibility Rules for 11 and 13 .
Most of the approaches I experienced at BSU was the professor(s) asking simple questions, giving students time to think and work in small groups, followed by some more key questions and think time. As a student this past summer at BSU, I appreciated that approach vs lots of information being thrown at me in a lecture. I hope you are able to learn a few new ways of helping students more effectively learn some of these ideas.