Mathematical Jeopardy
This is an excellent game for challenging students mathematical problem solving skills. It can be used in any area of the curriculum and at any age level. I have used it very successfully with multiplication and division problem solving with six year old students and here you see ten year old students solving perimeter and area problems.
You need 18 reallife word problems. The first 6 are basic and reinforce simple computation. Place a big 100 on the back of these. The second set, worth 200, involve a complication so some algebraic thinking is needed to solve them. The final set should be well above curriculum goals. They involve complicated reasoning and mixed operations to solve them, so they are valued at 300 points each. I usually find and adapt problems from a variety of teacher resources or websites. The students work in pairs with a buddy. They have complete freedom to decide on which problem to work on, but once chosen they solve it before moving to another one. They can't swap problems out if they don't like them. Those motivated by competition will naturally start comparing points. Other pairs just quietly enjoy their own success of solving problems. To get the points, the students must solve the problem in two distinctly different ways. I insist upon this as it forces kids to think outside the box of their own mind preferences. Flexible mathematical thinking  being able to solve problems in multiple ways is a continuing goal in all my mathematics classrooms. For this game I find mini whiteboards effective. Below the boys have agreed on the solution and are each showing it differently, one through a series of algorithms while the other is drawing the solution. 
I find a simple score card a handy record keeper. Once their two solutions have been evidenced by a teacher, we sign that the points have been awarded.
This lesson of 18 problems lasts for a good 90 minutes, depending on the complexity of the problems. Our math lessons last for 60 minutes and I find the students are very disappointed when the bell rings, so we often finish the next day. 
Introducing Multiplication  A Grade 1/Year 2 Unit
Above the chickens help students think in terms of ‘sets of’ or ‘groups of’. Again, I always introduce the concept first through oral stories only. After each child has worked out a way to solve the problem and the solution, I ask them how they could write it in a math sentence. Unless a child has been introduced to multiplication before, most children will naturally write a repeated addition sentence.
“There are four chickens. Each chicken eats four pieces of corn. How much corn needs to be scattered for the chickens to feed”. 4 + 4 + 4 + 4 = 16 or a skip counting sequence 4, 8, 12, 16 
Some will still simply count each corn kernel. This is fine if they are still able to represent the problem with a repeated addition sentence. If they cannot they need further practice in reframing the problem into groups. I always affirm the above responses as excellent mathematical thinking, and then I model and explain the multiplication sentence for them, as a new way to write the same idea. 4 groups of 4 = 16 or 4 X 4 = 16
If a child did use a multiplication sentence then I let them take the lead in explaining what their sentence means and why they wrote it that way. I always introduce the symbol ‘X’ at the same time as the concept ‘groups of’. Children readily understand the idea that we use symbols to write things quickly. Maths is not mysterious and it is not confusing. It is logical and sensible. We use  for take away, so naturally it is sensible to have a symbol for ‘groups of.’ 
I encourage the children to see problems in more than one way. It could be 15 groups of 2 or 6 groups of 5. Both ways of viewing the number of frogs above are equally valid. I think it is important for children displaying this type of flexible thinking to have practice explaining it and writing it mathematically. This clarifies their own understanding and provides a peer model. In the haunted house opposite, one child has decided to split the two 8s into 5s and 3s. When asked why, she said, “Its easier, ‘cause 5 + 5 is 10 and then the two 3s, 3 + 3 is 6. I can count 10 + 6, or I can count 10, 13, 16.”

At this stage we will also explore how multiplication can be flipped. It will be represented very differently, but the answer will stay the same. I will have the children work in partners. One partner will draw 2 haunted houses each with 8 ghosts while the other will draw 8 haunted houses each with 2 ghosts.

Working in rows is a very important step in helping children build thinking skills for heuristic problem solving. Rows are simply a more efficient way to organise our mathematical ideas than groups, especially when you are working with large numbers.

From sorting scattered groups into rows for easy counting, there is a natural progression to representing multiplication as arrays. This in turn will be a natural step into solving problems of area later on. I often have children work in pairs, one child will illustrate a sum while the partner will flip it.
6 X 3 = can be flipped to 3 X 6 = 
Once multiple strategies are known, I require the children to show all their working in at least 3 different ways for each problem. Here two children have used 4 different strategies successfully. This helps children to develop flexible thinking skills and to see mathematical problem solving as open, with multiple paths, rather than a single fixed algorithm. As a teacher, I can be confident that the children truly understand the concepts and mathematical principles when they can clearly demonstrate multiple paths to the answer. They can then make connections between the various representations, which is an important support for algebraic reasoning.
Once children are confidently showing clear understanding through multiple paths, I begin challenging them with problems that involve a missing factor. I do not introduce division at this point, but wait to see how the children use their multiplicative thinking and algebriac skills to present and solve the problem by themselves.
Once children are confidently showing clear understanding through multiple paths, I begin challenging them with problems that involve a missing factor. I do not introduce division at this point, but wait to see how the children use their multiplicative thinking and algebriac skills to present and solve the problem by themselves.
Memorization
I believe that rote learning still has a part to play in modern education. It all comes down to computational efficiency. Rote learned facts help us mentally solve day to day problems without having to carry a calculator and a bag of counters around with us. I do not believe that it is an either or situation. Both memorization and mental problem solving strategies are needed to be a competent mathematician in daily life. In year 2, I require that students memorize their 10s, 2s, and 5s. I also find that most children also competently grasp the x0s and x1s by the end of the unit as well. I do not find that I need to teach about the 0 factor or even the 1 factor: children understand them well after they have a solid understanding of X meaning groups of. 
I focus on each of the 10s, 2s, and the 5s for a whole week. Daily practice is encouraged through a nightly list of 25 problems and again each morning in school. This takes 3 weeks. In the fourth week, I review the 10s on Monday, the 2s Tuesday and the 5s on Wednesday. By Thursday, the children are reviewing all 3, and the 1s and 0s mixed in. This only takes the first 10 minutes of each lesson. I always present the problems in random order but with a line of skip counting on the top of the sheet for support. I have not found that I have ever needed the children to recite the tables, however we do skip count forwards and backwards constantly throughout the year. 