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The Ultimate Tool for Mathematical Reasoning
A Lesson in Mathematical Reasoning
A Subtle Change
Making Mathematical Reasoning A Homeschool Priority
In today's world, it seems there are thousands of skills we need to teach, but what if there was one skill we could teach that would be transferable to all subjects? Mathematical reasoning is that skill, and job market tells us it should be a homeschool priority.
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Why is Mathematical Reasoning Important?
Mathematical reasoning is a transferable thinking skill. It is not only important to math but to all subjects. Students who can reason about how information and ideas relate are also able to cultivate new ideas and discover new information. Knowing information is no longer useful. It is the ability to manipulate and use information to solve problems that will prove most valuable.
We have Google to find information, but Google can't synthesize information to create new ideas, let alone tell us how information can be valuable. That is still up to us, and with complex problems before us, we need innovators with the ability to reason and discover.
If there was one skill that we should prioritize, it is mathematical reasoning. Developing a strong memory is great but it won't stand to make an impact like one's ability to reason what is true. This ability to reason also helps us to know how to use information to solve problems. Because we are in the information age, the job market will continue to rely heavily upon those with strong reasoning skills.
The Ultimate Tool for Mathematical Reasoning
The Cuisenaire Rods are the ultimate tool for creating opportunities to develop mathematical reasoning. Students who play with Cuisenaire Rods perceive structures first before they see math facts. Because of this, they start to develop an understanding of how these structures work. This understanding gives them reasons to draw conclusions and test out their ideas which involves mathematical reasoning.
Developing mathematical reasoning begins with playing with math structures, and cuisenaire rods provide that visual for students to see those structures. This is the number one reason why Gattegno advocates using letters first to represent rods when teaching mathematical notation. Students who get caught up in numbers have a hard time seeing the structures. Manipulating rods help students to understand the structures and reason how manipulating the rods will impact the structures.
For example, light green plus light green equals a dark green. Seeing that adding two rods equals a larger rod allows the student to reason that the same structure might be true again. So the student tries two different rods to see if it equals a larger rod. Indeed, the student repeats their experiment over and over again, and they are able to reason that two rods of any length will always equal a large rod. They are noticing the big picture. With more play and more information, their understanding of math structures will grow and develop more depth. They will continue to reason if this is true, then this must be true, and so the student will test and prove out their ideas using Cuisenaire rods.
A Lesson in Mathematical Reasoning
Today’s exercise is perfect for extending that mathematical reason that has been developed through the literal work of Gattegno's textbook 1. Literal work is where the student is just building structures using the rods and using letters to write about those structures. Let's say you don't know anything about Gattegno or textbook 1. That is okay. You will still find this a useful activity for developing mathematical reasoning.
At some point during their interaction with Cuisenaire Rods, students decide the value of each Cuisenaire Rod. For older students, this happens more quickly. Without fail, the white rod gets assigned the value of one which then determines the rest of the rod values. Today's exercise challenges this perception of white equaling one, red equaling two and so on. We are going to change the value of the white rod, and students are going to use what they know to reason the new value of the rest of the rods.
Number sense is that fluid understanding of how numbers relate to each other, so when the value of the white rod no longer equals one, students are forced to really examine those relationships. Note: Students should probably be familiar with counting by tens and hundreds before diving into this lesson.
When I first began prepping for this activity, it did not occur to me that there was indeed purpose behind going from the value of white equals ten to 100 to 1/11th. As I did the activity, it soon became clear that the progression set the student up for the next lesson of changing the measuring rod.
This is a "flaw" to Gattengo's books. He assumes that you will do the activity yourself before presenting it to students. Today, "open and go" curriculums have allowed us to cut out a lot of prep time. We get to find out why we are doing what we are doing and the reasoning behind the progression. That is missing in Gattegno's textbooks more often than not and so many people miss the value in using Gattegno's textbooks (which are free and links are found in the resource section).
Today, I am going to set it up for you and hopefully, you can see the progression and use my tips to implement the lesson with your students with the same success. First, I began with the question "if white equals 10, what does red equal?"
The students might get stuck at first, not understanding your language and what you are asking. If this is the case, I suggest asking, “How many whites are in red?” This should be a familiar question that they can answer easily. If not, then the student is probably not ready for this activity.
DON'T RUSH
Give the student time to consider the implications of their answer and how it might help them determine the value of red. Success in math isn't based on how quickly one can find the answer. Excellent mathematicians are slow, deep thinkers that examine and ponder the details of the many relationships that exist simultaneously between structures and numbers.
Give students time to think. Walk away if you must. If a student continues to struggle, you can also encourage them to build a staircase and/or build trains of white rods to compare to each color rod. Notice that you are not giving them the answer or asking them lots of questions, but providing a gentle task that might help them see the answer for themselves. (If this unsuccessful, you might want to take a break, and return to literal activities.)
After some time, a light bulb should go off. Using a chart is helpful in allowing the student to see the progression, and I highly recommend using one throughout this exercise. Either you or the students record the new value of each rod in a table. As they progress to the orange rod, they should be rushing to answer, especially if they have counting by tens mastered. Next, I extended the lesson by changing the value of white to equal 100.
Next, I challenged them with white equals 4. They had to ponder a minute, but then they started to figure it out. One noticed, "Hey, we are counting by fours." Ahhhh, that is wonderful thing to notice.
A Subtle Change
The final challenge of the day was presented, white equals 1/11th. This is key step to the next lesson. It sets them up for changing the measuring rod. Now, we have been stopping at the orange rod, but this time ask them for the new value of orange plus a white rod. They should say eleven elevenths. Ask what is another way to see eleven elevenths? They should respond 1 whole. What they might not realize is that this means that 11 was the measuring rod. This will set them up for changing the measuring rod in the next lesson.
Side Note: What happens if they don't know that 11/11ths equals 1? I would return to fraction exercises in literal work, and abandon this lesson. They will not be able to bridge to the next lesson without this valuable understanding. Check out Sonya's post on literal fraction work for more help HERE and HERE.
During the lesson, I extended their observations by making problems and asking them what the problems would equal if white equals 10, 100, 4 or 1/11th. I created two-sided task cards that kept me on track in the progression and provided problems that were at the right level for their understanding.
The problems really help to open a door towards creating new, interesting expressions. That is the uniqueness of Cuisenaire rods and this activity. It creates lots of opportunity for the student to self-express themselves in mathematical notations. This activity can easily be extended by challenging the students to come up with their own values for white.
Deepening number sense and stretching mathematical reasoning continues when we change the measuring rod to red in the next lesson. We changed the measuring rod in that last task without them knowing it. Next time, we are going to be clear that we are changing the measuring rod.
Everything I have written should help you successfully implement this activity, but maybe you would love to have the task cards to make the process a little smoother. Go on over to my TPT store and pick up this activity which includes 40 double sided task cards, 10 "If, Then" Charts and 3 detailed lesson plans. There is a 48-hour sale making this product 50% off, so be sure to pick it up now.
Have you tried this activity or something similar? What is your favorite mathematical reasoning activity? Please share! Also, be sure to subscribe to my blog because I love delivering free gifts, tips and content right to your email box to make your math journey fun and easy.
