Tuesday, November 29, 2016

Interesting Ideas on Improving Mathematics Instruction on Select Topics

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.
Multiplying 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.

Monday, February 29, 2016

Mastery (Part 2)

Details of Practical Mastery
·         Mastery Quizzes (short quizzes)
o   Mastery quizzes can be done electronically like on Moodle or can be done on paper and pencil.  The key to the individual lesson mastery quiz is to keep them short.  A typical lesson mastery quiz will have at most 4 to 5 questions and be focused on a limited number of concepts.  At Byron our mastery quizzes and tests are paper/pencil since students do not have access to the internet when they take our state assessments every spring, the ACT, or their college placement tests, and we want to prepare our students for those assessments. 
o   One of the challenges of mastery quizzes is making similar but different versions of a quiz.  You do not want to just change the numbers because students will memorize the process versus understanding the concepts.
o   After a student takes a mastery quiz, you want to provide feedback on that quiz ASAP, ideally within a minute or two since they are still emotionally invested in the quiz. 
o   Grade mastery quizzes on a 1, 2, 3, or 4 scale.  4 is perfect, 3 is proficient, 2 is some understanding of the material with room to grow, 1 is lots of opportunity to improve.
o   If a student does not do well on a quiz, for example a 2 out of 4, you still need to celebrate with the students on what they know and help focus their energies toward what they don’t know.  This explicit identification of students know and do not know is empowering as it helps them see success and focus efforts appropriately.
o   Encourage students to strive for a 4 on all mastery quizzes.  When students are pushed to get 4’s on all mastery quizzes, they do a lot better on the unit tests assuming your mastery quizzes are aligned to your unit tests
o   Students are not allowed to retake a mastery quiz on the same day they just went over their mastery quiz.  Students need a sleep cycle between going over their mastery quiz and retaking it.  We also recommended that students learn and practice the material one day then wait until the next day to take a mastery quiz.  This encourages a deeper learning and longer retention of the material. 
o   Whether you are quizzing online or paper pencil mastery quizzes, be sure to avoid the line of students waiting for you.  Students may be in line to get a mastery check, have you grade a mastery check, go over the mastery check with you, or ask a question on an assignment.  But if students are standing in line, they are wasting their class-time.  You will need to figure out a process that works for you to avoid the line that can easily occur.
·         You will need to change the physical setup of your room and have one area specifically for the mastery quizzes with another area for group work or individual work.  We have one row of desks set up for students who want to take mastery quizzes then tables or groups of desks setup for students to work on learning and practicing concepts.
·         Provide a pacing guide but encourage students work ahead since they will come across harder sections that take more time and/or may be gone someday.
·         In class encourage students to focus and make good use of every minute.  For many students that means if they focus and make good use of their time then they will not need to watch any videos or complete any math problems outside of class while others will need to do some homework for math to stay on pace.
·         Encourage all students to retake mastery quizzes on test review days, including students who got all fours on their mastery quizzes.  Supporting and encouraging long-term retention of concepts is a key goal.
·         On final review days, students are allowed to retake unit tests to improve their score, again demonstrating retention and mastery of concepts.
Benefits of Mastery
·         The culture of a mastery classroom is different.  Students have more of a growth mindset or the “I can do it if I work at it”, “not yet”, or “Try Fail, Try Fail, Try Again then Success” mindset.  In a non-mastery classroom if a student failed an assessment, it reinforces the student’s idea that “I am not good at math.” But in a mastery classroom if  students fail an assessment, they have the attitude that they are going to go back and relearn the material so that I can master the concepts.
·         There is a big change in the conversations within the mastery classroom.  The student-to-student conversations or the student-to-teacher conversations are more focused on learning the material rather than just getting the assignment done like is often true in a non-mastery classroom.
·         Students view the homework problems as a tool for learning the material versus something they have to get done even if they do not learn anything from it.
·         Students in a mastery classroom take more pride and ownership of their own learning.
·         When a student does fail your course, you have an exact record of what they know and what they do not know.  So it is relatively easy to get their grade up to a C if they want to.  This can easily be done by having them attend your “summer school” for only a few days and focus specifically on the concepts they do not understand versus needing to retake the whole course next school year or in full summer school session.