Making Connections Creates Long-Term Learning
by Bill Reed
I was working with a school and the teachers started talking about the NCAA Basketball tournament. I quickly zoned out as with my basketball prowess I have no interest in basketball at all. The entire discussion became background noise to me. I could not tell you a single thing that was discussed during that entire conversation. If the topic would have been about the current NHL standings and the upcoming Stanley Cup Playoffs my ears would have tuned in immediately and I would have been laser focused on every word that was being said. I have played hockey since I was six years old. I am not a professional player (and never have been), but I like watching and playing the sport and I understand the rules and how it is played. I can appreciate the amazing talent some players have with what they can do on the ice!
I realized that this is the same reaction many of our students have when math teachers start talking about math for their daily lesson. First, they are not interested in the topic. Second, they may or may not understand the rules, intricacies, and beauty of the subject. Finally, they do not see any connections to want is being taught or meant to be understood. Teachers must make some connections to the math they are teaching so the students at least give some thought and focus to what is being presented.
This leads me to the most important part of a teacher’s lesson. The hook! Teachers, especially math teachers, must take some time to plan a very pointed and focused hook for their students. They must utilize a variety of methods to engage the students’ focus so they will “tune in to” the lesson the teacher is about to teach. If the teacher does not engage the students from the start of the lesson using fun and interesting methods, everything the teachers says become Charlie Brown’s teacher voice. You know; “Wah, wah, wah wah wah, wah wah! …”
Ways to Engage
There are a variety of ways teachers can engage students. They can use humor. They can use games and puzzles. They can use personal anecdotes. They can peak the students’ interests or challenge them to use what they already know to start a lesson. Whatever method the teacher chooses, they need to relate it to their students, so they make some sort of a connection. If this is done, the student will at least listen and hear what is being said. It does not mean they will internalize and learn the lesson being taught. It does mean you have a chance to get the student learning the material being taught and, with practice it will lead to mastery.
We must make the connections students know and understand. I used to start many of my Algebra 1 lessons with taking the students back to elementary school and the lessons they learned in Grades 1 – 6. I would tell them I would be teaching them many of the same basic math principles they already learned and knew how to do. This at least got my foot in the door relating it to skills they have already done and processes they were familiar with that they had done in the past.
Take for instance the Distributive property. I would start with a simple problem very familiar for the students like 8 x 12. I would have them do it the way most students were taught using the multiplication algorithm. I would then break it apart for them using the Distributive Property 8(10 + 2). We then would do a few simple examples they knew and could easily do. After I had them “hooked”, I would turn the discussion to a more abstract level. I would ask them how them how they would split up 8 x 37. Most students would have some success splitting up 37 into (30+7). We talked about how, with the Distribute Property, you never had to do more than a single digit times a single digit multiplication. I told students they only had to know their times tables up to 9 x 9. I would then challenge the students to prove me wrong, knowing this would lead them into the next steps I wanted them to take.
Students would of course go to problems like 7 x 132. We broke that into 7(100+30+2) which means they could easily get 700+210+14. Most students could add those three numbers in their head and get 924. Of course, students LOVE to trip up and mess up the teacher so they would get cute and asked about 23 x 54. I would say “Glad you asked that!”. Let’s tear this apart. 23 x 54 can be split up into (20+3)*(50+4). We can then just make that two different Distribute Properties that can be combined. 20(50+4) and 3(50+4). Those are both simple easy single digit multiplication problems. 1000 + 80 for the first one and 150 + 12 for the second one. I then showed them if we combine them, it would be 1000 + 80 + 150 + 12 which equals 1242. Students could easily add these numbers and we would verify our answers using a calculator or the standard multiplication algorithm.
Distributive Properties to Algebra Problems
I would them take it to the next level and turn these problems into true Algebra problems with variables. I circled back to the very first problem and asked what it would look like if I asked them instead of multiplying 8 x 12, I said 8 x some two-digit number ending in 2. It could be 8 x 12, 8 x 22, 8 x 32, 8 x 42, 8 x 52, 8 x 62, 8 x 72, 8 x 82, or 8 x 92. But I told them I didn’t want to work 9 problems. I only wanted to work 1 problem. I explained mathematics are lazy they NEVER want to do more work or write more than they had to to get their information shown!
Funny, that may have been the most relatable part of my lesson to many on the students, but I digress.
I told them we could use a variable to represent the tens digit and say 8 x n2 but that is confusing since a number next to a variable is also multiplication. I showed them we would write it 8(n+2) just like we had split up the 12 earlier. This made sense to them because they had just seen it earlier in the class. I asked them to work the problem just like we had been working all the problems. It would be 8*n + 16. I shared with the students that this is how their calculator and computer programs work problems like this. The programmer has to do the problems in general so the method would work no matter what number was chosen.
We would revisit this lesson many times throughout the year. We had already touched on it with the example of 23 x 54. Did you realize that double digit multiplication is really the infamous F.O.I.L.? This is why I do not like using or teaching F.O.I.L.! I always teach expanded Distributive Property. It will work for any polynomial time any polynomial not just a binomial time a binomial. 23 x 54 can be split up into (20+3)*(50+4). We can then just make that two different Distribute Properties that can be combined. 20(50+4) and 3(50+4). I then showed them if we combine them, it would be 1000 + 80 + 150 + 12 which equals 1242.
Binomials and Factoring Quadratics
I would use these same methods to teach the two special cases for multiplying binomials and factoring quadratics. I would start by asking students to do a simple easy multiplication problem. Something like 68 x 72. The students would use a calculator or the multiplication algorithm and get 4896. I showed them I worked the problem as 68 x 72 = 4900 – 4 = 4896. (When I was much younger and in better practice, I could get the answer quicker than they could on a calculator). I asked them what I had done and turned it into a puzzle or solvable mystery for them. After some discussion amongst themselves they usually could relate the 702 = 4900 and 22 = 4 but they usually did not get why the minus sign. I showed them that Difference of Two Squares they were learning states (a+b)(a–b) = a2 –- b2. 68 x 72 could be written (70–2)(70+2) = 4900 – 4 = 4896. Can you do 49 x 51 much easier now in your head?
Likewise, I would have students do a problem like 572. Again, the students would use a calculator or the multiplication algorithm and get 3249. I showed them I worked the problem as 572 = 2500 + 700 + 49 = 3249. I asked them what I had done and turned it into a puzzle or solvable mystery for them. After some discussion amongst themselves they usually could relate the 502 = 2500 and 72 = 44 and most of the time they did come up with 7 x 50 was 350 and double 350 is 700 but they usually did not get why they had to double the 360 or why it was all addition. I showed them that Perfect Square Trinomial they were learning states (a+b)2 = a2 + 2ab + b2. By starting with what they knew and relating it to things we had already done it gave the lesson meaning and value.
Ask For Help From Your Students
When you really think about it, there is very little if anything you teach in Algebra 1, until maybe Quadratics, that students haven’t seen before. You should be able to relate what you are teaching to things students have seen and done before. If you are at a loss for ways to relate what you are teaching, ask the students to help you. You can use students relating the math they are learning to something they have done before or to an actual common application of the mathematics they are learning as a replacement for a missed assignment, a missed problem on a quiz or test. I believe students looking up, relating and/or researching the information they are learning in class is just as important, and maybe even more important, than any one assignment, problem on a quiz, or a problem on a test. Instead of extra credit points for bringing in supplies, not using the restroom during class time, or other non-math related items why not relate extra credit with doing more and going further with the math the students are learning? I think this is a great way for students to help their grade by making the math they are learning more relatable. Just an idea.
Overall, the more relatable you can make what you are teaching, the more the students will connect and see a reason for the math you are helping them learn. Research and expert articles show just how important making these connections are for students. There are many articles that directly link the students’ feelings about and retention of the mathematics they were learning to how relatable the math they were learning was to the student. This should be no surprise to anyone in education. This is basic human nature. If we understand and relate to something we find it interesting or at least tolerable. If we see no purpose, no connection, or does not relate to us, we tune out and forget what is being said as quickly as possible. We all do it. I am guessing every person reading this can think of a time where they have done exactly what I am describing here.
Finally, the one thing I want you to consider when you think about your next lesson is that the lesson you are teaching is one of many lessons the students will be getting that day. What makes your lesson any more than just noise or minutiae to the student? The more you can relate your lesson to your students, the more they will remember and engage in the lesson you are teaching.
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