What do we look for in a mathematics curriculum?
First – let’s think for a moment about the traditional math curriculum. Is it working?
Not really – why are so many kids still struggling with math? Why do so many families have to get their children outside help for their children? Why do we have to repeat the same concepts over and over and over again each year?
What is in the traditional curriculum? Children learn from many different goals. Although I understand the desire to cover all of mathematics and give children exposure to multiple areas of mathematics – I think we should focus on the basics and overall number sense before we worry about teaching children all the vocabulary needed for Geometry. Why does a 4th grader need to know the name “Obtuse” angle when they still haven’t grasped multiplication and divisibility? Most of the Geometry concepts we teach can really be taught later while we use our time more effectively to build number sense among children. Each year, children need to re-learn the Geometry vocabulary we throw at them because we don’t actually “use” Geometry, we just insist that they need to know the names of angles, triangles, and which angles are congruent when 2 parallel lines are cut by a transversal. We should teach all of this Geometry when students have learned enough mathematics that they can use and see the importance of Geometry.
Concepts like perimeter and area can be taught and used as an application to addition and multiplication but so much of the rest of the Geometry we teach to elementary and middle school children is just done to justify that we are teaching “all” topics in math instead of teaching what is needed at that particular time.
The only Geometry topics that I think the elementary child needs to know are: what different degree measures mean (what does it mean to turn 90 degrees), perimeter, area, and basic shapes.
What about the number sense? We teach basic facts but we do not try to build number sense with children. Even if you ask a child this second grade goal questions, “Write 3 different ways to represent 250.” Does the student even understand the question? If they respond 250 ones or 25 tens – do they really get what that means? No, because we don’t teach things in the right order. If you want a child to understand that 250 is the same as 25 tens – you should have introduced multiplication by then. Yet, multiplication isn’t taught until 3rd grade and our students are supposed to answer this question in 1st and 2nd grade. They learn to repeat what they are told but don’t really understand at a deeper level – how can they? They haven’t learned all the needed pieces to full grasp it.
The curriculum needs to stop thinking, “Multiplication is taught after addition and subtraction.” Multiplication links into a lot of concepts we teach younger children. When it links in, we should introduce it. If we do that, multiplication will have a lot more meaning.
Progressive Math is a curriculum that is designed to make the links as they come up. It has no rules of that say “A must be taught before B.” The curriculum guides itself. If a so called “later topic” (such as multiplication) presents itself conceptually in a unit taught to a kindergartener (think 4+4 means 2 sets of 4 means 2X4) then the concept is taught. Math makes more sense since topics are linked to other topics. It is a chain of links together rather than our current curriculum which is a bunch of disjoint sets. We teach kids to count by 5′s – what concepts link to that? Telling time, multiplication, counting coins, etc. All these things link to counting by 5′s and therefore we connect them. This is something that is missing in current curricula.
Another problem when traditional math is taught, is that a concept is taught and then forgotten about. Maybe the student will remember, maybe not. Progressive Math constantly reviews old topics so students are constantly getting practice of the older material so they don’t forget as well as moving forward. How many times has a child taken a unit test and once it was over, forgot everything. If you gave that student the same unit test 2 months later, how would they do? If they really understand, link concepts, and have regular practice, they should do fine.
Another issue we hope to resolve is the steps involved from getting from point A to point D. Many traditional curriculum teach all the steps A to B, B to C and C to D together in one “step,” this is fine for some students who can make that leap but what about those that can’t and get lost or even if they get it but find it “hard.” What if, instead, we gave mini-lessons that taught students to just do step A to B. When they had some practice, we teach B to C, and then finally from C to D. Now we can do the A to D jump and all students are right there with us because we took the time to break down the intermediate steps. Here is an example:
Goal: Teach time to the 5 minute interval.
Traditional Lesson: Students have been taught time to the hour, half-hour, and now (a year later) must go to 5 minute interval. They must see a clock and be able to say the time is 2:35.
Progressive Lesson:
1. Student learns to count by 5′s.
2. Student learns how to associate the numbers on the clock that the big hand points to as values when one counts by 5′s. (They only focus on the big hand and the concept of counting by 5′s, there is no telling time here – it is just a basic lesson on counting by 5′s but using the clock numbers as our counters.)
3. Students learn how to tell what time the small hard is referencing.
4. Student takes step #3 and combines it with step #2 and can tell the time to the 5 minute interval.
The progressive lesson is a more natural approach. It links things the student already knows (counting by 5′s) to telling time. The traditional approach has students memorize what a clock looks like when it is on the hour or half hour. Next the student has to make a large jump from the half hour time to the 5 minute interval with out any bridge.
If we linked all our mathematics together, if we encouraged overall number sense, and we provided intermediate steps our children would learn faster and better.
The traditional approach also relies a lot on memorization. Students are told to memorize their addition facts. How many kids are still counting on their fingers in second, third, fourth, and later for addition? I have high school students I see who still use their fingers. That is the only strategy they have! They don’t have any number sense because no one ever taught them number sense. Instead of memorizing 6+7=13, a student with number sense who has learned to link concepts can use a variety of approaches to solve the problem in their head without counting on their fingers. For some reason most kids are able to learn their doubles easily – so why not rely on the doubles. If a child knows that 6 + 6 = 12 but they don’t know 6 + 7 they can apply number sense and know that 6 + 7 must be just one more than 6 + 6, therefore it is 13. Our curriculum doesn’t teach strategies, they don’t teach or build number sense so students have to memorize a whole bunch of unrelated pieces of information rather than using what they do remember and pulling from that to get what they don’t have memorized.
This concept of number sense goes far beyond just addition or multiplication facts. It serves as backbone of how to think logically and do what mathematicians do all the time – use prior knowledge and link that together to find answers to things they don’t know. This is a skill that helps students through all their years of math and in real life. Our schools, our curricula, just don’t take advantage of that. Instead we teach third graders the name “Obtuse angle.”
We apply this idea of Progressive teaching to all of our mathematics curriculum. Visit us at Apex-Math to see what we have to offer.
